# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 2 Elliptical Characteristic of Exoatmospheric Targets  
The entire trajectory of a ballistic target can be divided into three basic portions: boost, ballistic (coast, midcourse, or exoatmospheric), and reentry [47]. Among these, the boost and reentry phases are relatively short compared with the Earth's radius, and take place only in close vicinity of Earth. On the contrary, the ballistic phase is an exoatmospheric, free-flight motion, continuing until the atmosphere is reached again. Usually, a typical intercontinental ballistic missile (ICBM) can last about 30 min in the midcourse, which constitutes the main flight time and range of a ballistic target's motion. Therefore, the defense radar must detect and track this type of target earlier. In particular, it is expected that the target will be detected and tracked during the entire exoatmospheric phase. Under these circumstances, the defense system will have more time to intercept a target and also evaluate its performance of the engagement. During exoatmospheric flight, the predominant force acting on the target is Earth's gravity, while the effect of atmospheric drag, Earth non-sphericity, and the gravitational forces due to other celestial bodies can be ignored. Under these assumptions, the dynamics (motion) models of the target are governed by the laws of [PERSON] (1953). Since the target has a negligible mass relative to the Earth, the gravitational acceleration of the target is the solution of a so-called restricted two-body problem, obtained by [PERSON]'s inverse-square gravity law as  
\[\mathbf{a}_{G}(\mathbf{r})=-\frac{\mu}{r^{2}}\mathbf{u}_{\mathbf{r}}=-\frac{ \mu}{r^{3}}\mathbf{r} \tag{1}\]  
where \(\mathbf{r}\) is the vector from the Earth's center to the target, \(r\triangleq\left|\mathbf{r}\right|\) is its length, \(\mathbf{u}_{\mathbf{r}}\triangleq\mathbf{r}/\mathbf{r}\) is the unit vector in the direction of \(\mathbf{r}\), and \(\mu=3.986005\times 10^{14}\,\mathrm{m}^{3}/\mathrm{s}^{2}\) is the Earth gravitational constant. Using Equation (1) combined with the conservation laws of mechanical energy and moment of momentum, it can be demonstrated that the trajectory follows an elliptical orbit in essence. For a comprehensive treatment of the theories and solutions to the two-body problem, the reader is referred to (1953). To illustrate this, a new coordinate system (CS) is set up (we denote it as THIS-CS), and the target-Earth-radar geometry is shown in Figure 1. In this scenario, the curve \(\mathbf{\Gamma}\) denotes the target trajectory in the exoatmospheric phase, with A the thrust cutoff or burnout point, C the apogee, and B the reentry point. The inside sphere \(\mathbf{\Omega}_{\mathrm{R}}\) denotes the Earth's surface, while the outside sphere denotes the atmospheric boundary (in this paper, we simply assume that this boundary coincides with the turnoff and reentry points. The THIS-CS \(\mathrm{O}_{\mathrm{e}}-xyz\) is a right-handed inertial system, with the origin \(\mathrm{O}_{\mathrm{e}}\) at Earth's center, axis \(\mathrm{O}_{\mathrm{e}}x\) pointing in the AB direction, axis \(\mathrm{O}_{\mathrm{e}}y\) pointing in the \(\mathrm{O}_{\mathrm{e}}\mathrm{C}\) direction, and axis \(\mathrm{O}_{\mathrm{e}}z\) perpendicular to the orbit plane. In this user-defined CS, the elliptical orbit of the target in the exoatmospheric phase can be simply formulated as  
\[\mathbf{\Gamma}:\ \frac{x^{2}}{b^{2}}+\frac{\left[y-\left(a^{2}-b^{2}\right)^{1 /2}\right]^{2}}{a^{2}}=1,\ \left|x\right|\leq\left(r_{\mathrm{e}}+H_{\mathrm{C}}\right)\sin(\theta/2),\ y\geq 0,\ z=0 \tag{2}\]  
where \(a\) and \(b\) are the semi-major and semi-minor axes of the ellipse, respectively, i.e., \(a=\mathrm{OC}\) and \(b=\mathrm{OD}\). Note that wherein \(\mathrm{O}\) is the real ellipse center, while the Earth center is only a focus, with half focal length \(\mathrm{O}\mathrm{O}_{\mathrm{e}}=\sqrt{a^{2}-b^{2}}\).