# Initial Identification of Thrust and Orbit Elements for Continuous Thrust Spacecraft in Circular Orbit

^{1}

^{2}

^{*}

## Abstract

**:**

^{−7}m/s

^{2}, and that of normal acceleration is around 3 × 10

^{−6}m/s

^{2}; the accuracy of the semi-major axis is 350 m, and the accuracy of inclination is 0.095°. The method is applicable to the preliminary identification of thrust and orbit elements for circular orbit continuous thrust spacecraft and can provide reliable initial values for the subsequent precision orbit determination of such spacecraft.

## 1. Introduction

## 2. Modelling of Orbit Motion

#### 2.1. Thrust and Orbit Elements

- (a)
- The semi-major axis variation is related only to the tangential acceleration and not to the orbit plane normal acceleration.
- (b)
- The inclination variation is related to the normal acceleration and the spacecraft velocity. There is a fixed mathematical relationship between the spacecraft velocity and the half-length axis, so the change in inclination is related to both the normal acceleration in the expression and the normal acceleration (the term affecting the change in the semi-major axis).

#### 2.2. Approximate Solution of the Perturbation Equations

#### 2.2.1. Continuous Thrust Motion Control Equations

#### 2.2.2. Analytical Model for Semi-Major Axis

#### 2.2.3. Analytical Model of Inclination

_{0}is the initial velocity of the spacecraft. Integrate the above equation:

#### 2.2.4. Analytical Model of Argument of Latitude

#### 2.2.5. Analysis of Perturbation Effects

_{D}is the drag coefficient, S

_{d}is the windward surface area, m is the spacecraft mass, S/m is the surface-to-mass ratio, ρ is the atmospheric density, v is the spacecraft velocity, and

**v**is the spacecraft velocity vector. Recent atmospheric density data published by NASA were used [22]. The values of the parameters involved in Equation (28) are taken: ρ = 1.38 × 10

^{−16}g/cm

^{3}at an altitude of 600 km, C

_{D}= 2.2, S/m = 0.02, and v = 7.56 × 10

^{3}m/s. The atmospheric drag acceleration is calculated to be 1.736 × 10

^{−7}m/s

^{2}in the opposite direction of the spacecraft velocity. From the performance of existing space electric thrusters [23], the acceleration produced by the thruster is generally in the range of 10

^{−3}–10

^{−5}m/s

^{2}, which will be 2 to 4 orders of magnitude larger than the acceleration produced by atmospheric drag. The acceleration generated by atmospheric drag in the opposite direction of the velocity can be considered when identifying the thrust parameters of a continuously thrusting spacecraft to compensate for the desired tangential thrust acceleration.

_{R}is the solar pressure coefficient, S

_{R}is the equivalent area, m is the spacecraft mass, p

_{s}is the solar pressure constant at the location of the spacecraft, and v

_{s}is the sun exposure coefficient, which is 1 when the spacecraft is illuminated by sunlight, and conversely 0.

**i**

_{s}characterizes the direction of the solar pressure perturbation force and is the unit vector of the spacecraft to the center of the sun. The literature [24] shows that the acceleration produced by solar pressure on a spacecraft is on the order of 10

^{−7}and hardly affects the thrust acceleration produced by the electric thrusters.

_{0}= (r

_{0}, i

_{0}, Ω

_{0}, u

_{0}), the state of the spacecraft at time t can be expressed by the following equation.

## 3. Transformation of Orbit Elements

#### 3.1. Semi-Major Axis Transformation

_{0}and tangential acceleration ${\overline{F}}_{T}$ can affect each parameter. The first equation in Equation (2) implies that the initial semi-major axis and tangential acceleration can be solved by studying the change law of the semi-major axis, which can simplify the process of solving the other parameters in the following. Under the J2000.2 equatorial reference system (geocentric inertial coordinate system), the radar observation data are transformed into spacecraft semi-major axis data. The spatial relationship between the geocenter, the station, and the spacecraft is shown in Figure 6.

**r**shows the position of the spacecraft,

**R**shows the position of the station, and

**ρ**shows the position of the spacecraft relative to the station.

#### 3.2. Inclination Transformation

**O**) and the equatorial plane (

**E**) in the xOy plane, which can be expressed as z = 0.

**l**

_{0}and

**l**formed by the intersection of the initial orbit plane

_{i}**L**

_{0}and the plane

**L**with the equatorial plane

_{i}**E**, respectively, and the angular difference is ΔΩ

_{i}. Knowing that the initial RAAN is Ω

_{0}, the node line l0 can be expressed as

_{i}is

**Ω**, and the intersection line

_{i}= Ω_{0}+ ΔΩ_{i}**l**can be expressed as

_{i}_{0}is the position of the continuous thrust spacecraft at the initial moment t

_{0}, and P

_{i}denotes the position of the spacecraft at the moment t

_{i}; L

_{0}is the initial orbit plane, L

_{i}is the orbit plane at the moment t

_{i}, and the expressions for P

_{0}:(x

_{0}, y

_{0}, z

_{0}) and P

_{i}:(x

_{i}, y

_{i}, z

_{i}) are as follows:

_{0}and the straight line OP

_{0}are both on the initial orbit plane L

_{0}and are not parallel when u ≠ 0. The normal vector h

_{0}to the plane L

_{0}is

**e**is defined as the normal vector of the equatorial plane

**E**.

**E**is the xOy plane in the space Cartesian coordinate system, so

**e**= (0,0,1). The angle made by the plane

**L**

_{0}and the equatorial plane

**E**is the initial orbit inclination i

_{0}.

_{i}at time t

_{i}is

#### 3.3. Argument of Latitude Transformation

_{i}and the node line l

_{i}is the argument of latitude u.

## 4. Thrust and Orbit Elements Solving Methods

^{2}, the statistical properties of the observations are as follows:

_{0}, tangential thrust acceleration ${\overline{F}}_{T}$, initial inclination i

_{0}, initial RAAN Ω

_{0}, normal thrust acceleration ${\overline{F}}_{N}$, and initial argument of latitude u

_{0}can be obtained sequentially.

## 5. Simulation Analysis

^{−4}m/s

^{2}. At this time, the yaw angle is set to be 30°, the tangential acceleration is 1.966 × 10

^{−4}m/s

^{2}, and the normal acceleration is 1.135 × 10

^{−4}m/s

^{2}. All simulation conditions are set as shown in Table 1.

^{−4}[20], so the tangential acceleration and normal acceleration can be taken as 0, which is the initial value when simulation analysis is performed. The solution results are shown in Table 2.

^{−7}m/s

^{2}, and the relative error of the normal acceleration solution is about 3 × 10

^{−6}m/s

^{2}. The error of the semi-major axis is about 0.35 km, the error of solving the inclination is about 0.095°, and the error of the RAAN and argument of latitude are about 0.74° and 0.83°, respectively. Figure 9 shows the solution error of tangential acceleration, normal acceleration, semi-major axis, inclination, RAAN, and argument of latitude.

## 6. Conclusions

- (1)
- The adoption of the analytical orbit model can reflect the relationship between the elements more clearly, and, at the same time, simplify the calculation and quickly solve the problem.
- (2)
- The decoupling of tangential thrust and normal thrust parameters can simplify the solving process, and the solving accuracy of tangential acceleration and normal acceleration can reach the order of 10
^{−6}and 10^{−5}m/s^{2}, which can be directly applied to engineering practice. - (3)
- In the process of solving the RAAN and argument of latitude, only the long-term effect of the spacecraft by the J2 term perturbation is considered, and its short-term effect with the thrust acceleration is not considered, so the solution accuracy is lower, but it can be used as the initial value to be substituted into the next step of the precise orbit determination.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Project | Parameter |
---|---|

Initial epoch time | 2023.04.02 04:46:39 UTC |

Initial orbit | $[r,i,\Omega ,u]=[6933.5345\mathrm{k}\mathrm{m},65.011\xb0,14.372\xb0,28.728\xb0]$ |

Initial acceleration | $[{\stackrel{\u02c9}{F}}_{T},{\stackrel{\u02c9}{F}}_{N}]=[0.0001966\mathrm{m}/{\mathrm{s}}^{2},0.0001135\mathrm{m}/{\mathrm{s}}^{2}]$ |

Station coordinates | [−2852.90 km, 3399.95 km, 4565.25 km] |

Three body gravity | Sun, moon, and major planets: JPL DE405 |

Observation error (Gaussian distribution) | $[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3}]=[0.03\mathrm{km},\frac{0.1}{\mathrm{cos}(El)}\xb0,0.1\xb0]$ |

Tide | Solid tide: IERS Conventions 2003 |

Non-spherical gravitational field | Gravity model: EGM2008 Degree:21; order:21 |

Relativity | IERS Conventions 2003 |

Solar pressure | Shadow model: Dual Cone Light pressure coefficient: 1.00 Area–mass ratio: 0.02 m ^{2}/kg |

Atmospheric drag | Density model: Jacchia–Roberts Drag coefficient: 2.20 Area–mass ratio: 0.02 m ^{2}/kg |

Orbit Parameter | Simulation Value | Solution Result | Error |
---|---|---|---|

Tangential acceleration (m/s^{2}) | 0.0001966 | 0.0001961 | 0.0000005 |

Normal acceleration (m/s^{2}) | 0.0001135 | 0.0001106 | 0.0000029 |

Semi-major axis (km) | 6933.534519 | 6933.885632 | 0.351 |

Inclination (°) | 65.011 | 64.916 | 0.095 |

RAAN (°) | 14.372 | 13.629 | 0.743 |

Argument of latitude (°) | 28.728 | 27.894 | 0.834 |

Thrust | 1 N | 500 mN | 100 mN | 20 mN | 5 mN | 1 mN |
---|---|---|---|---|---|---|

Acceleration (m/s^{2}) | 2 × 10^{−3} | 1 × 10^{−3} | 2 × 10^{−4} | 4 × 10^{−5} | 1 × 10^{−5} | 2 × 10^{−6} |

Simulation tangential acceleration (m/s^{2}) | 1 × 10^{−3} | 5 × 10^{−4} | 1 × 10^{−4} | 2 × 10^{−5} | 5 × 10^{−6} | 1 × 10^{−6} |

Solution tangential acceleration (m/s^{2}) | 1.000 × 10^{−3} | 5.000 × 10^{−4} | 0.999 × 10^{−4} | 2.033 × 10^{−5} | 4.645 × 10^{−6} | 0.486 × 10^{−6} |

Simulation normal acceleration (m/s^{2}) | 1.732 × 10^{−3} | 8.660 × 10^{−4} | 1.732 × 10^{−4} | 3.464 × 10^{−5} | 8.660 × 10^{−6} | 1.732 × 10^{−6} |

Solution normal acceleration (m/s^{2}) | 1.738 × 10^{−3} | 8.624 × 10^{−4} | 1.622 × 10^{−4} | 2.682 × 10^{−5} | 5.890 × 10^{−6} | 1.223 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Zhao, S.; Tao, X.; Li, Z.
Initial Identification of Thrust and Orbit Elements for Continuous Thrust Spacecraft in Circular Orbit. *Aerospace* **2023**, *10*, 1012.
https://doi.org/10.3390/aerospace10121012

**AMA Style**

Zhao S, Tao X, Li Z.
Initial Identification of Thrust and Orbit Elements for Continuous Thrust Spacecraft in Circular Orbit. *Aerospace*. 2023; 10(12):1012.
https://doi.org/10.3390/aerospace10121012

**Chicago/Turabian Style**

Zhao, Shuailong, Xuefeng Tao, and Zhi Li.
2023. "Initial Identification of Thrust and Orbit Elements for Continuous Thrust Spacecraft in Circular Orbit" *Aerospace* 10, no. 12: 1012.
https://doi.org/10.3390/aerospace10121012