Particle energisation in a collapsing magnetic trap model : the relativistic regime

The authors acknowledge financial support by the UK’s Science and Technology Facilities Council through a Doctoral Training Grant (SEO) and Consolidated Grant ST/K000950/1 (SEO and TN).Context. In solar flares, a large number of charged particles is accelerated to high energies. By which physical processes this is achieved is one of the main open problems in solar physics. It has been suggested that during a flare, regions of the rapidly relaxing magnetic field can form a collapsing magnetic trap (CMT) and that this trap may contribute to particle energisation. Aims. In this Research Note we focus on a particular analytical CMT model based on kinematic magnetohydrodynamics. Previous investigations of particle acceleration for this CMT model focused on the non-relativistic energy regime. It is the specific aim of this Research Note to extend the previous work to relativistic particle energies. Methods. Particle orbits were calculated numerically using the relativistic guiding centre equations. We also calculated particle orbits using the non-relativistic guiding centre equations for comparison. Results. For mildly relativistic energies the relativistic and non-relativistic particle orbits mainly agree well, but clear deviations are seen for higher energies. In particular, the final particle energies obtained from the relativistic calculations are systematically lower than the energies reached from the corresponding non-relativistic calculations, and the mirror points of the relativistic orbits are systematically higher than for the corresponding non-relativistic orbits. Conclusions. While the overall behaviour of particle orbits in CMTs does not differ qualitatively when using the relativistic guiding centre equations, there are a few systematic quantitative differences between relativistic and non-relativistic particle dynamics.Publisher PDFPeer reviewe

Writhe formulas and antipodal points in plectonemic DNA configurations

The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double-helix, these numbers are not always straightforward to compute, though a simplified formula exists. We examine the range of applicability of this widely-used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works

Singular inextensible limit in the vibrations of post-buckled rods: analytical derivation and role of boundary conditions

In-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible planar Kirchhoff elastic rod under large displacements and rotations. Equilibrium configurations and vibrations around these configurations are computed analytically in the incipient post-buckling regime. Of particular interest is the variation of the first mode frequency as the load is increased through the buckling threshold. The loading type is found to have a crucial importance as the first mode frequency is shown to behave singularly in the zero thickness limit in case of prescribed axial displacement, whereas a regular behavior is found in the case of prescribed axial load

Frequency jumps in the planar vibrations of an elastic beam

The small amplitude transverse vibrations of an elastic beam clamped at both extremities are studied. The beam is modeled as an extensible, shearable planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is the variation of mode frequencies as the load is increased through the buckling threshold. While for some modes there is no qualitative changes in the mode frequencies, other modes experience rapid variations after the buckling threshold. For slender beams, these variations become stiffer, eventually resulting in a discontinuous jump of frequency at buckling, in the limit of inextensible, unshearable beams

Elasticity and electrostatics of plectonemic DNA

We present a self-contained theory for the mechanical response of DNA in single molecule experiments. Our model is based on a 1D continuum description of the DNA molecule and accounts both for its elasticity and for DNA-DNA electrostatic interactions. We consider the classical loading geometry used in experiments where one end of the molecule is attached to a substrate and the other one is pulled by a tensile force and twisted by a given number of turns. We focus on configurations relevant to the limit of a large number of turns, which are made up of two phases, one with linear DNA and the other one with superhelical DNA. The model takes into account thermal fluctuations in the linear phase and electrostatic interactions in the superhelical phase. The values of the torsional stress, of the supercoiling radius and angle, and key features of the experimental extension-rotation curves, namely the slope of the linear region and thermal buckling threshold, are predicted. They are found in good agreement with experimental data.Comment: 19 pages and 6 figure

Integrals of motion and the shape of the attractor for the Lorenz model

In this paper, we consider three-dimensional dynamical systems, as for example the Lorenz model. For these systems, we introduce a method for obtaining families of two-dimensional surfaces such that trajectories cross each surface of the family in the same direction. For obtaining these surfaces, we are guided by the integrals of motion that exist for particular values of the parameters of the system. Nonetheless families of surfaces are obtained for arbitrary values of these parameters. Only a bounded region of the phase space is not filled by these surfaces. The global attractor of the system must be contained in this region. In this way, we obtain information on the shape and location of the global attractor. These results are more restrictive than similar bounds that have been recently found by the method of Lyapunov functions.Comment: 17 pages,12 figures. PACS numbers : 05.45.+b / 02.30.Hq Accepted for publication in Physics Letters A. e-mails : [email protected] & [email protected]

On the logarithmic probability that a random integral ideal is A\mathscr A-free

This extends a theorem of Davenport and Erd\"os on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and provide a formula for the logarithmic density of the set of so-called $\mathscr A$-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set $\mathscr A$.Comment: 9 pages, to appear in S. Ferenczi, J. Ku{\l}aga-Przymus and M. Lema\'nczyk (eds.), Chowla's conjecture: from the Liouville function to the M\"obius function, Lecture Notes in Math., Springe