赤道平面J2束缚轨道研究

对近地航天器有显著影响的J2项摄动势函数是轴对称的.而当航天器的轨道倾角为零时,J2项势函数由轴对称退化为球对称.本文从能量的角度来研究近地航天器J2束缚轨道特性.通过引入＂势能井＂的概念,证明了J2项势函数作用下,赤道平面内存在两类束缚轨道：环形束缚轨道与圆形封闭轨道.根据总能量与有效势的关系,分析了形成这两类J2束缚轨道的条件.利用球对称势的周期性理论,借助椭圆积分推导了环形J2轨道径向周期、周向周期和向径的表达式,以及圆形J2轨道的周期和角速度.分析了圆形封闭轨道的稳定性,以及受到小扰动后向径的演化规律.最后探讨了这两类J2束缚轨道潜在的应用.

Anatomy of J2 bounded orbits in equatorial plane

The function of J2 perturbative potential working significantly on spacecraft is axially symmetric. While the orbital inclination equals to zero, the J2 potential reduces to spherically symmetric potential. The characteristic of J2 bounded orbits of low altitude is anatomized from the energy point of view. The existence of two types of bounded orbits with J2 effect named annular orbits and circular orbits in equatorial plane is demonstrated by means of the concept of a ＂Potential Energy Well＂. Under what condition will the J2 bounded orbits be generated is studied by discussing the relationship between the total energy and effective potential. Then the condition is transformed to a discriminant A of a cubic function. For orbits with fixed angular momentum, if A 〈 0, viz, the curve of effective energy varying with time has three intersections with the total energy, two of which are great than the radius of the Earth, the spacecraft will be trapped in the well and orbits are quasi-periodic, namely annular orbits. If A = 0, only one intersection exists and the motion is uniform circular. For annular orbits, the radial period, azimuthal period and radius are expressed in terms of elliptic integrals applying periodic theory of spherically symmetric potential. For circular orbits, the expression of its period and angular velocity is derived. Then the stability is analyzed via the small disturbance linearization method, and evolution of radius with perturbation is derived thereafter. In the end, a method for generating natural bounded relative orbits over infinite time is illustrated, by means of locating two spacecrafts, the chief and the deputy, on J2 bounded orbits respectively. According to the stability analysis in sect. 2.2, the geometry of relative orbits is stable.