标题： 带有正负点质量的牛顿限制三体引力问题 显示英文标题

标题： Newtonian restricted three-body gravitational problem with positive and negative point masses

摘要： 涉及一个正的一次点质量$m_+$和一个负的二次点质量$m_-$在圆轨道上，以及一个正或负的三次点质量$m_3$与$m_+ > |m_-| \gg |m_3|$的牛顿限制三体问题被解决了。 对于$m_3$找到了五个拉格朗日点，其中三个与$m_+$和$m_-$共面，另外两个则不共面，这是来自$m_-$的引力排斥的微妙结果。 所有拉格朗日点都是线性不稳定的，除了在区间$m_+ \gtrsim 8.4 |m_-|$中的一个点，该点是线性稳定的，并且与$m_+$和$m_-$在同一直线上。 显示英文摘要

摘要： The Newtonian restricted three-body problem involving a positive primary point mass, $m_+$, and a negative secondary point mass, $m_-$, in a circular orbit, and a positive or negative tertiary point mass, $m_3$, with $m_+ > |m_-| \gg |m_3|$, is solved. Five Lagrange points are found for $m_3$, three of which are coplanar with $m_+$ and $m_-$, and two of which are not, a subtle consequence of the gravitational repulsion from $m_-$. All Lagrange points are linearly unstable, except for one point in the regime $m_+ \gtrsim 8.4 |m_-|$, which is linearly stable and collinear with $m_+$ and $m_-$.