It is well known that a planar central configuration of the \(n\)-body problem gives rise to a solution where each particle moves in a Keplerian orbit with a common eccentricity \(\mathfrak{e}\in[0,1)\). We call this solution an elliptic relative equilibrium (ERE for short). Since each particle of the ERE is always in the same plane, it is natural to regard it as a planar \(n\)-body problem. But in practical applications, it is more meaningful to consider the ERE as a spatial \(n\)-body problem (i. e., each particle belongs to \(\mathbb{R}^{3}\)). In this paper, as a spatial \(n\)-body problem, we first decompose the linear system of ERE into two parts, the planar and the spatial part. Following the Meyer – Schmidt coordinate [19], we give an expression for the spatial part and further obtain a rigorous analytical method to study the linear stability of the spatial part by the Maslov-type index theory. As an application, we obtain stability results for some classical ERE, including the elliptic Lagrangian solution, the Euler solution and the \(1+n\)-gon solution.

众所周知，\(n\)体问题的平面中心配置产生了一个解决方案，其中每个粒子在具有共同偏心率的开普勒轨道上移动\(\mathfrak{e}\in[0,1) \)。我们将此解称为椭圆相对平衡（简称 ERE）。由于 ERE 的每个粒子总是在同一平面上，因此很自然地将其视为平面\(n\)体问题。但在实际应用中，将 ERE 视为空间\(n\)体问题（即每个粒子属于\(\mathbb{R}^{3}\) ）更有意义。在本文中，作为一个空间\(n\)体问题，我们首先将ERE线性系统分解为平面和空间两部分。继Meyer-Schmidt坐标[19]之后，我们给出了空间部分的表达式，并进一步通过Maslov型指数理论得到了研究空间部分线性稳定性的严格解析方法。作为应用，我们获得了一些经典 ERE 的稳定性结果，包括椭圆拉格朗日解、欧拉解和\(1+n\)边形解。