General relativistic numerical scheme for quasi equilibrium states of irrotational binary neutron star systems

我们的研究目标是建立一个强大的数值格式来求解广义相对论中的双中子星系统，并研究双中子星系统的演化。随后，我们将能够为双中子星合并模拟提供初始模型。我们需要开发一个强大的数值格式来求解广义相对论中的双中子星系统。我们成功地制定了一项新方案，其中下列两种方案发挥了重要作用。(1)首先将球坐标原点移至原坐标原点的下半部分区域。(2)其次，通过引入一个新的空间来消除轨道运动旋转轴上的不良行为，在这个新空间中，平坦空间中的拉普拉斯算子可以由原始爱因斯坦方程的非线性椭圆算子推导出来。在这个新的空间里，爱因斯坦方程通过适当的格林函数被转换成积分方程。这两个新公式使我们得到了一个鲁棒的数值格式，可以很容易地给出双中子星的准平衡解。应用这一新格式，我们求解了具有较高刚度和较强引力的双中子星结构。具有恒定重子质量的双中子星的演化序列显示如下结果。如果我们使用几个现实的状态方程，演化将在不稳定开始时结束，而使用软状态方程的多向星将达到接触状态而不会遭受不稳定。

Our goal in this investigation is to develop a powerful numerical scheme to solve binary neutron star systems in general relativity and to study evolutions of binary neutron star systems. Subsequently we will be able to offer initial models to binary neutron star merging simulations.We need to develop a robust numerical scheme to solve binary neutron star systems in general relativity. We have succeeded in developing a new scheme in which the following two formulations play essential roles. (1) First, the origin of the spherical coordinates are shifted to the lower half region of the original coordinate origin. (2) Second, the bad behavior on the rotational axis of the orbital motion can be eliminate by introducing a new space in which a Laplacian operator in the flat space can be deduced from the nonlinear elliptical operators of the original Einstein equations. In that new space, the Einstein equations are transformed into integral equations by making use of the appropriate Green functions. These two new formulations allow us to get a robust numerical scheme which can give quasiequilibrium solutions of binary neutron stars easily.By applying this new scheme, we have solved structures of binary neutron stars with relatively high stiffness and strong gravity. Evolutionary sequences of binary neutron stars with constant baryon masses show the following results. If we use several realistic equations of state, evolutions will end at the onset of instability, while polytropic stars with soft equations of state would reach contact states without suffering from instability.