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

我们的研究目标是发展一个强有力的数值方法来求解广义相对论中的中子星星系统，并研究中子星星系统的演化。随后我们将能够提供初始模型来模拟中子星星并合，我们需要发展一个稳健的数值方案来求解广义相对论中的中子星星系统。我们成功地制定了一个新的计划，其中以下两个公式发挥了重要作用。(1)首先，将球坐标的原点移动到原始坐标原点的下半区域。(2)第二，通过引入一个新的空间，由原Einstein方程的非线性椭圆算子导出平坦空间中的Laplacian算子，可以消除轨道运动在旋转轴上的不良行为。在这个新的空间中，利用适当的绿色函数，将爱因斯坦方程转化为积分方程。这两种新的格式使我们得到了一种能方便地给出中子星拟平衡解的稳健的数值格式，并应用这种新格式求解了具有较高刚度和较强引力的中子星结构。重子质量恒定的中子星双星演化序列表明：如果我们使用几个现实的状态方程，演化将在不稳定性开始时结束，而具有软状态方程的多变恒星将在不受不稳定性影响的情况下达到接触状态。

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.