Development of numerical scheme for equilibrium structures and their stability analysis of strongly magnetized rotating stars

在这项研究中，我们成功地发展了强磁化旋转恒星轴对称和静止结构的新公式，并根据该公式获得了新的数值代码。通过数值计算，我们得到了磁化旋转恒星的多种平衡构型。我们还开发了一种新的稳定性公式，目前正在构建一个用于稳定性分析的数值代码。关于轴对称和静止磁化旋转恒星，从20世纪50年代开始就有许多研究者进行了研究。然而，在考虑强磁场的情况下，得到的平衡构型都是只有极向磁场的构型。然而，只有极向场的构型被认为是不稳定的。因此，需要构造具有极向和环向磁场的平衡构型。为建立具有极向和环向磁场的平衡结构的新的数值格式，寻找更大电流密度的表达式是一个关键问题。在本研究中，我们得到了电流密度的一般表达式：j=(κ+φ‘v)H+φ Ω +ρ(μ+Ω’v)e_φ，其中j、H、Ω、ρ和e_φ分别为电流密度、磁场、涡量、质量密度和旋转方向上的单位矢量。这里，κ, φ， μ和Ω是向量势的φ分量的任意函数。利用该表达式，我们成功地得到了(1)强磁化且均匀旋转的恒星，(2)强磁化且差异旋转的恒星，以及(3)具有极向和环向磁场的经向循环的强磁化旋转恒星。从这些平衡构型得到的最重要的结果与轴对称定态一般构型的磁场分布有关，即磁场的扭环面分布是磁平衡星的最一般构型。它们由类偶极极场和绕几何环面绕轴对称轴旋转的环面场组成。需要强调的是，这种磁场的扭曲环面结构是由Brauithwaite和Spruit通过动力学模拟得到的。因此，这些扭曲的环面结构被认为是静止轴对称恒星磁场的稳定和一般分布。少

In this investigation we have succeeded in developing a new formulation of structures of axisymmetric and stationary structures of strongly magnetized rotating stars and obtaining a new numerical code according to the formulation. By using the numerical code we have obtained many kinds of equilibrium configurations of magnetized rotating stars. We also have developed a new stability formulation and are now constructing a numerical code for the stability analysis.Concerning axisymmetric and stationary magnetized rotating stars, many researchers have studied them from 1950's. However, only equilibrium configurations obtained have been those for configurations with only poloidal magnetic fields by taking the strong fields into account. Nevertheless, configurations with poloidal fields only have been considered to be unstable. Thus it has been required to construct equilibrium configurations with both poloidal and toroidal magnetic fields. It is a crucial point to find an expression of the … More electric current density in order to develop a new numerical scheme for equilibrium configurations with poloidal and toroidal magnetic fields.In this investigation, we have found the general expression for the electric current density as follows :j=(κ+φ'v)H+φω+ρ(μ+Ω'v)e_φwhere j, H, ω, ρ and, e_φ are the current density, the magnetic field, the vorticity, the mass density, the unit vector in the rotational direction, respectively. Here, κ, φ, μ and Ω are arbitrary functions of the φ component of the vector potential.By using this expression, we have succeeded in obtaining (1) strongly magnetized and uniformly rotating stars, (2) strongly magnetized and differentially rotating stars and (3) strongly magnetized rotating stars with meridional circulations with both poloidal and toroidal magnetic fields. The most important result obtained from these equilibrium configurations is connected with the magnetic field distributions of general configurations for axisymmetric stationary states : i.e. twisted torus distributions of the magnetic fields are the most general configurations for the magnetic equilibrium stars. They consist of the dipole-like poloidal fields and the toroidal fields which spiral around geometrical tori around an axisymmetric axis. It should be stressed that this kind of twisted torus structures of the magnetic fields have been obtained by dynamical simulations by Brauithwaite and Spruit. Thus these twisted torus structures are considered to be stable and general distributions of the magnetic fields for stationary axisymmetric stars. Less