论文标题
RTBP中的$ n $ exjection-Collision轨道上的广义分析结果。分叉分析
Generalised analytical results on $n$-ejection-collision orbits in the RTBP. Analysis of bifurcations
论文作者
论文摘要
In the planar circular restricted three-body problem and for any value of the mass parameter $μ\in (0,1)$ and $n\ge 1$, we prove the existence of four families of $n$-ejection-collision ($n$-EC) orbits, that is, orbits where the particle ejects from a primary, reaches $n$ maxima in the distance with respect to it and finally collides with the primary.这样的EC轨道具有$ c =3μ+ln^{2/3}(1-μ)^{2/3} $的jacobi常数的值,其中$ l> 0 $足够大,但独立于$μ$和$ n $。为了证明这一最佳结果,我们考虑Levi-Civita的转换,以一种使用Ad Hoc Small参数在配置平面中使用适当的比例,并以前已应用了时间。该结果改善了先前的工作,其中$ n $ -ec轨道的存在是在质量参数$μ> 0 $足够小时所说的。在本文中,考虑了$μ\ in(0,1)$和$ n \ ge 1 $的任何可能值。此外,对于$ c $的降低值,出现了一些分叉,这些分叉首先是数字研究的,然后讨论了分叉值$ c $的明确表达式。最后,还描述了$ n $ -ec轨道存在$μ\至1 $的详细分析。在自然的方式中,希尔的问题出现了。对于这个问题,我们证明了四个$ N $ -EC轨道的家族的分析结果,从数字上讲,我们描述了它们以及出现的分叉。
In the planar circular restricted three-body problem and for any value of the mass parameter $μ\in (0,1)$ and $n\ge 1$, we prove the existence of four families of $n$-ejection-collision ($n$-EC) orbits, that is, orbits where the particle ejects from a primary, reaches $n$ maxima in the distance with respect to it and finally collides with the primary. Such EC orbits have a value of the Jacobi constant of the form $C=3μ+Ln^{2/3}(1-μ)^{2/3}$, where $L>0$ is big enough but independent of $μ$ and $n$. In order to prove this optimal result, we consider Levi-Civita's transformation to regularize the collision with one primary and a perturbative approach using an ad hoc small parameter once a suitable scale in the configuration plane and time has previously been applied. This result improves a previous work where the existence of the $n$-EC orbits was stated when the mass parameter $μ>0$ was small enough. In this paper, any possible value of $μ\in (0,1)$ and $n\ge 1$ is considered. Moreover, for decreasing values of $C$, there appear some bifurcations which are first numerically investigated and afterwards explicit expressions for the approximation of the bifurcation values of $C$ are discussed. Finally, a detailed analysis of the existence of $n$-EC orbits when $μ\to 1$ is also described. In a natural way Hill's problem shows up. For this problem, we prove an analytical result on the existence of four families of $n$-EC orbits and numerically we describe them as well as the appearing bifurcations.