## 3 Single-Arc Least-Squares Orbit Determination  
Initial orbit determination usually considers two-body dynamic models, the results of which provide initial values for precision orbit determination without the use of optimal estimation theory. For radar observations, the commonly used algorithms for initial orbit determination are the Gibbs method and the Herrick-Gibbs method. For measured angular data only, commonly used algorithms for initial orbit determination include the Laplace method, Gaussian method, and Gooding method.  
Initial orbit determination is usually carried out using single-arc measurements. However, individual observation arcs are usually short (low-orbit target arcs are usually only a few minutes to ten minutes) and orbit determination accuracy is limited. The process of orbital change under continuous thrust involves low thrust and a slow rate of change in orbital elements. If the orbit determination accuracy is too low, it will lead to a large relative error in the thrust acceleration calculation. In order to improve the accuracy of orbit determination for single-arc segments, the least-squares method of orbit determination considering the J2 term perturbation is used.  
Define the \(n\) sets of noisy measurements of distance, azimuth, and pitch angle obtained by ground-based radar observation equipment as  
\[\widetilde{z}_{i}=(t_{i},\widetilde{p}_{i},\widetilde{A}_{i},\widetilde{E}_{i}), i=1,2,\cdots,n \tag{2}\]  
\[\widetilde{\mathbf{Z}}=[\widetilde{z}_{1},\widetilde{z}_{2},\cdots,\widetilde{z}_ {n}] \tag{3}\]  
Let the truth value of the measurement sequence be  
\[\begin{array}{c}-\\ \widetilde{\mathbf{Z}}=[\overline{z}_{1},\overline{z}_{2},\cdots,\overline{z}_{n} ]\end{array} \tag{4}\]  
\[\overline{z}_{i}=(t_{i},\overline{p}_{i^{\prime}},\overline{A}_{i},\overline{ E}_{i}),i=1,2,\cdots,n \tag{5}\]  
Assuming that the observation errors follow a normal distribution with a mean squared error of \(\sigma^{2}\), the statistical properties of the observations are  
\[\begin{cases}\widetilde{p}_{i}\sim\mathcal{N}(\overline{p}_{i^{\prime}}, \sigma_{1}^{2})\\ \widetilde{A}_{i}\sim\mathcal{N}(\overline{A}_{i},\sigma_{2}^{2})\\ \widetilde{E}_{i}\sim\mathcal{N}(\overline{E}_{i},\sigma_{3}^{2})\end{cases} \tag{6}\]  
Define the state quantities of the dynamic system as \(\mathbf{X}(t)=[\mathbf{r};\dot{\mathbf{r}}]\), where  
\[\mathbf{r}=\begin{bmatrix}x&y&z\end{bmatrix}^{T},\dot{\mathbf{r}}=\begin{bmatrix}\dot{x }&\dot{y}&\dot{z}\end{bmatrix}^{T}\]  
The systematic observation model is  
\[\widetilde{z}_{i}=\mathbf{h}[\mathbf{X}(t_{i})]+\mathbf{v}_{i},i=1,2,\cdots,n \tag{7}\]  
where \(v_{i}\) is the systematic observation noise.  
This gives  
\[\mathbf{H}(\mathbf{X}(t_{0}))=\begin{bmatrix}\mathbf{h}(\mathbf{X}(t_{1})),\mathbf{h}(\mathbf{X}(t_{2} )),\cdots,\mathbf{h}(\mathbf{X}(t_{n}))\end{bmatrix} \tag{8}\]  
The least-squares orbit determination problem is described as solving for the optimal estimate of the orbit state \(\mathbf{X}(t_{0})\) at a given calendar moment \(t_{0}\) using a least-squares criterion that minimizes the sum of squares of the weighted residuals of the theoretical and actual observations.  
\[\min_{\mathbf{X}_{0}}\!\!\!/(\mathbf{X}(t_{0}))=\frac{1}{2}[\widetilde{\mathbf{Z}}-\mathbf{H} (\mathbf{X}(t_{0}))]^{T}\mathbf{W}[\widetilde{\mathbf{Z}}-\mathbf{H}(\mathbf{X}(t_{0}))] \tag{9}\]  
where a matrix of weights \(\mathbf{W}\) is given to account for the accuracy of different observations, usually weighted by the inverse of the error in the observations \(\sigma\).