# Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem
## 2 Description of the Problem  
The single-impulse flyby two co-orbital spacecraft problem is described in this paper as using a spacecraft with an orbital altitude lower than the target orbit to fly by two spacecraft at different positions in the target orbit using a single-impulse maneuver.  
There are three primary spacecraft in this problem: a chase vehicle named \(SC_{0}\), and two target spacecraft named \(ST_{1}\) and \(ST_{2}\). There are four significant times: the initial time \(t_{s}\), the impulse time \(t_{0}\), the time the chaser intercepts the first target \(t_{1}\), and the time the chaser intercepts the second target \(t_{2}\). There are three main orbits: the chaser's orbit \(O_{0}\), the target's orbit \(O_{1}\), and the chaser's transfer orbit \(O_{2}\).  
All variables in this article have two subscripts. The first indicates the satellite number, with 0 representing the chaser, 1 representing target 1, and 2 representing target 2. The second subscript represents the time, such as \(t_{s}\) or \(t_{1}\). Each spacecraft's position at four main times is depicted in Figure 1. The problem of the single-impulse flyby of two co-orbital targets can be described as follows. First, at the time \(t_{0}\), the chaser \(SC_{0}\) transfers from the orbit \(O_{0}\) to the orbit \(O_{2}\). The chaser \(SC_{0}\) then flies by the first target at the time \(t_{1}\), at which point their positions are \(\mathbf{r}_{0,t_{1}}\) and \(\mathbf{r}_{1,t_{1}}\), respectively. Finally, at the time \(t_{2}\), the chaser \(SC_{0}\) flies by the target \(ST_{2}\), whose positions are \(\mathbf{r}_{0,t_{2}}\) and \(\mathbf{r}_{2,t_{2}}\). Note that the chaser may perform a flyby of the target \(ST_{2}\) first. Therefore, the central equation between the positions in this problem can be written as  
\[\left\{\begin{array}{l}\mathbf{r}_{0,t_{1}}=\mathbf{r}_{1,t_{1}}\\ \mathbf{r}_{0,t_{2}}=\mathbf{r}_{2,t_{2}}\end{array}\right.or\left\{\begin{array} []{l}\mathbf{r}_{0,t_{1}}=\mathbf{r}_{2,t_{1}}\\ \mathbf{r}_{0,t_{2}}=\mathbf{r}_{1,t_{2}}\end{array}\right. \tag{1}\]  
The problem may be further abstracted as the following mathematical problem after describing the known conditions and unknown variables. Given the orbital elements of three spacecraft at the initial time \(t_{s}\), the transfer time \(t_{0}\) and the velocity change vector \(\Delta\mathbf{V}\) necessary for the chaser \(SC_{0}\) should be found to fly by the targets \(ST_{1}\) and \(ST_{2}\).  
According to the problem description, there are six unknown values in this problem: the impulse time \(t_{0}\), the three-dimensional velocity change vector \(\Delta\mathbf{V}\), and the flyby times \(t_{1}\) and \(t_{2}\). Simultaneously, the connections between these variables may be used to simplify them. For example, there is a Lambert problem between the unknown variables \(t_{0}\)\(t_{1}\) and \(\Delta\mathbf{V}\). The change vector \(\Delta\mathbf{V}\) and the orbital elements of the transfer orbit may be uniquely calculated for a given \(t_{0}\) and \(t_{1}\). The intersection of the transfer and target orbit is therefore determined, and the position of each of the two intersections is the position where the chase performs a flyby of each of the two targets. Accordingly, given the time \(t_{0}\) and the time \(t_{1}\), the second flyby position can likewise be predicted--hence, the time \(t_{2}\). In this method, the problem of a single-impulse flyby of two co-orbital targets is reduced from a six-dimensional problem to a two-dimensional problem and then to a one-dimensional problem for a given \(t_{0}\). Therefore, in the next section, a theoretical derivation of this method is presented, followed by its one-dimensional constraint relation equation.  
Figure 1: Geometric interpretation of a single-impulse flyby two co-orbital spacecraft problem.