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Bistatic Radar Observations Correlation of LEO Satellites Considering J_{2} Perturbation

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## Abstract

**:**

## 1. Introduction

## 2. Initial Orbit Determination for Tracklets Observed by Bistatic Radar

- angles observed by the receiving station, usually azimuth and elevation $(A,E)$ in topocentric horizontal coordinate system for radars;
- the sum of ranges by the transmitting station and receiving station, $\rho ={\rho}_{T}+{\rho}_{R}$. ${\mathbf{\rho}}_{\mathit{T}}={\rho}_{T}\widehat{{\mathbf{\rho}}_{\mathit{T}}}$ and ${\mathbf{\rho}}_{\mathit{R}}={\rho}_{R}\widehat{{\mathbf{\rho}}_{\mathit{R}}}$, $\widehat{{\mathbf{\rho}}_{\mathit{T}}}$ and $\widehat{{\mathbf{\rho}}_{\mathit{R}}}$ are the unit vector of ${\mathbf{\rho}}_{\mathit{T}}$ and ${\mathbf{\rho}}_{\mathit{R}}$, ${\rho}_{T}$ and ${\rho}_{R}$ are the length of ${\mathbf{\rho}}_{\mathit{T}}$ and ${\mathbf{\rho}}_{\mathit{R}}$. Sum of ranges $\rho $ can be measured directly, but ${\rho}_{T}$ and ${\rho}_{R}$ are unknown.

- Angles only: Since a series of angle observations is sufficient for IOD, a sum of ranges can be temporarily put aside. With Equation (1),$$\widehat{{\mathbf{\rho}}_{\mathit{R}}}\times \mathit{r}=\widehat{{\mathbf{\rho}}_{\mathit{R}}}\times {\mathit{r}}_{\mathit{R}},$$${\rho}_{R}$ from the receiving station to the space debris can be eliminated; therefore, only azimuth and elevation observed by the receiving station are used. The defect of this approach is that low-accuracy measurements are used, and high-accuracy measurements are rejected.
- ${\mathbf{\rho}}_{\mathit{R}}$ calculation: Since ${\mathit{r}}_{\mathit{R}\mathbf{2}\mathit{T}}={\mathit{r}}_{\mathit{T}}-{\mathit{r}}_{\mathit{R}}$, $\widehat{{\mathbf{\rho}}_{\mathit{R}}}$ and $\mathbf{\rho}$ are known with a simple geometric calculation, ${\mathbf{\rho}}_{\mathit{R}}$ can be obtained. Then, the position of space debris can be calculated. The problem of this approach is that the errors of angles are transferred to ${\mathbf{\rho}}_{\mathit{R}}$.

## 3. Orbit Improvement with Weighted Least Square Method

#### 3.1. Weighted Least-Squares Method

#### 3.2. Effect of Weight

## 4. Correlation Considering ${J}_{2}$ Perturbation

#### 4.1. Orbit Propagation with ${J}_{2}$ Perturbation

#### 4.2. Correlation of Tracklets

- $\Delta {\rho}_{R}$ the error produced by the receiving station;
- $\Delta {\rho}_{T}$ the error produced by the transmitting station;
- $\Delta {\rho}_{s}$ the systematic error between the receiving station and the transmitting station.

## 5. Discussion

- (1)
- the performance of orbit determination with WLSM for a single bistatic radar tracklet;
- (2)
- the effect of orbital elements’ accuracy and prediction duration on $\Delta \rho $.

#### 5.1. Accuracy of Orbit Determination

- Orbit improvement with WLSM is indispensable. Since the error of the estimated orbit elements by IOD was too large, the number of miscorrelations would grow rapidly as the interval between tracklets increased. At the same time, the systematic bias of the estimated semimajor axis would render the orbit propagation wrong.
- As mentioned in Section 4.2, an empirical error of the estimated orbit is needed to calculate ($\Delta {A}_{k}^{j}$, $\Delta {E}_{k}^{j}$, $\Delta {\rho}_{k}^{j}$) in Equation (26). From Figure 5, the empirical error can be obtained.

#### 5.2. Variation in and Evolution of $\Delta \rho $

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Simulated Stations | Latitude (deg) | Longitude (deg) | Height (m) |
---|---|---|---|

Transmitting station | 30 | 108 | 0 |

Receiving station | 30 | 105 | 0 |

Semi-Major Axis (m) | Eccentricity | Inclination (deg) | RAAN (deg) |
---|---|---|---|

6,878,137.0 | 0.001 | 60.0 | 60.0 |

Subject | Content |
---|---|

Orbital elements | Two-Line-Element (TLE) |

Dynamic model | sgp4 |

Minimal height threshold | 200 km |

Maximal height threshold | 1700 km |

Minimal time span | 20 s |

Maximal time span | 300 s |

Mean time span | 120 s |

$\sigma $ of azimuth noise | $0.1\xb0$ |

$\sigma $ of elevation noise | $0.1\xb0$ |

$\sigma $ of $\rho $ noise | 50 m |

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**MDPI and ACS Style**

Huyan, Z.; Jiang, Y.; Li, H.; Ma, P.; Zhang, D.
Bistatic Radar Observations Correlation of LEO Satellites Considering *J*_{2} Perturbation. *Mathematics* **2022**, *10*, 2197.
https://doi.org/10.3390/math10132197

**AMA Style**

Huyan Z, Jiang Y, Li H, Ma P, Zhang D.
Bistatic Radar Observations Correlation of LEO Satellites Considering *J*_{2} Perturbation. *Mathematics*. 2022; 10(13):2197.
https://doi.org/10.3390/math10132197

**Chicago/Turabian Style**

Huyan, Zongbo, Yu Jiang, Hengnian Li, Pengbin Ma, and Dapeng Zhang.
2022. "Bistatic Radar Observations Correlation of LEO Satellites Considering *J*_{2} Perturbation" *Mathematics* 10, no. 13: 2197.
https://doi.org/10.3390/math10132197