# Approximations for Secular Variation Maxima of Classical Orbital Elements under Low Thrust

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## Abstract

**:**

## 1. Introduction

## 2. Gauss’s Variational Equations and Simplifying Assumptions

#### 2.1. Gauss’s Variational Equations for Classical Orbital Elements

#### 2.2. Simplifying Assumptions

- The fuel consumption is small and ignored due to the low magnitude of the thrust compared to the total mass of the spacecraft. Consequently, the magnitude of the propulsive acceleration ${a}_{max}$ becomes constant because of the constant low-thrust magnitude ${T}_{max}$. Thus, the following equation holds$$\parallel \mathit{a}\parallel =\frac{{T}_{max}}{m}={a}_{max}=\mathrm{const}$$The thrust magnitude ${T}_{max}$ is modeled as a function of the maximum thruster input power ${P}_{max}$ and specific impulse ${I}_{\mathrm{sp}}$ as [40]$${T}_{max}=\frac{2\eta {P}_{max}}{{I}_{\mathrm{sp}}{g}_{0}}$$
- The variation of the true anomaly caused by the three components of $\mathit{a}$ can be ignored because its maximum ${a}_{max}$ is much smaller than the central gravitational acceleration [34].$$\frac{\mathrm{d}f}{\mathrm{d}t}\approx \frac{h}{{r}^{2}}$$In Equation (9), the approximation of the Equation (6) can be derived when the effect on $\mathrm{d}f/\mathrm{d}t$ caused by the low thrust is small enough compared to the term $h/{r}^{2}$ (a similar approximation can also obtained in [36,41]).Divided by Equation (9), Gauss’s variational equations of classical orbital elements are transformed into the following differential equations in terms of the true anomaly$$\frac{\mathrm{d}a}{\mathrm{d}f}=\frac{2{a}^{2}{r}^{2}v}{\mu h}{a}_{t}$$$$\frac{\mathrm{d}e}{\mathrm{d}f}=\frac{{r}^{2}}{hv}\left[2(e+cosf){a}_{t}-\frac{r}{a}sinf{a}_{n}\right]$$$$\frac{\mathrm{d}i}{\mathrm{d}f}=\frac{{r}^{3}cos\theta}{{h}^{2}}{a}_{h}$$$$\frac{\mathrm{d}\Omega}{\mathrm{d}f}=\frac{{r}^{3}sin\theta}{{h}^{2}sini}{a}_{h}$$$$\frac{\mathrm{d}\omega}{\mathrm{d}f}=\frac{{r}^{2}}{hev}\left[2sinf{a}_{t}+\left(2e+\frac{r}{a}cosf\right){a}_{n}\right]-\frac{{r}^{3}sin\theta cosi}{{h}^{2}sini}{a}_{h}$$

## 3. Secular Variation Maximum of Single Classical Orbital Element

#### 3.1. Secular Variation Maximum of Semi-Major Axis

Algorithm 1: Iterative algorithm. |

Input:Initial orbital elements ${a}_{0},{e}_{0},{i}_{0},{\Omega}_{0}$, and ${\omega}_{0}$ For each loop iteration of each subsection:- 1.
- Calculate the variation maxima $\Delta {\mathit{x}}_{N}$ over one orbital revolution
- 2.
- Add the variations of the orbital elements ${\mathit{x}}_{N}$ = $\Delta {\mathit{x}}_{N}$ + ${\mathit{x}}_{N-1}$
Stopping conditions:The time of flight reaches the given value. Output:Approximations for the variation of the orbital elements ${\mathit{x}}_{N}$ |

#### 3.2. Secular Variation Maximum of Eccentricity

#### 3.3. Secular Variation Maximum of Inclination

#### 3.3.1. Strategy 1

#### 3.3.2. Strategy 2

#### 3.4. Secular Variation Maximum of Right Ascension of the Ascending Node

#### 3.5. Secular Variation Maximum of Argument of Periapsis

## 4. Numerical Simulations

#### 4.1. Simulations for Variation Maximum of Each Orbital Element

#### 4.2. Estimation of the Velocity Increment

## 5. Discussion

## 6. Conclusions

## 7. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | |

a | Semi-major axis |

e | Eccentricity |

i | Inclination |

$\Omega $ | Right ascension of ascending node |

$\omega $ | Argument of periapsis |

f | True anomaly |

h | Magnitude of specific angular momentum |

$\mu $ | Gravitational constant |

$\mathit{a}$ | Propulsive acceleration vector |

$\Delta \mathit{x}$ | Variation group of the classical orbital elements over one orbital revolution |

${\mathit{x}}_{N}$ | Values of the orbital elements after N orbital revolutions |

${T}_{max}$ | Thrust magnitude |

${R}_{E}$ | Mean equatorial radius of the Earth |

${J}_{2}$ | Second order zonal harmonic of the Earth’s gravitational potential |

$\Delta V$ | Velocity increment |

$\mathit{X}$ | Group of the orbital elements |

$\mathit{u}$ | Thrust vector |

u | Engine thrust ratio |

$\alpha $ | Unit vector of thrust direction |

$\mathrm{ToF}$ | Time of flight |

$\lambda $ | Lagrange multiplier associated with state, i.e., costate |

H | Hamiltonian |

AU | Astronomical unit |

$\mathit{\Phi}$ | Combination of shooting functions |

m | Instantaneous mass of spacecraft |

$\beta $ | Out-of-plane (yaw) steering angle |

$\alpha $ | In-plane thrust-steering angle |

Subscripts | |

$LT$ | Low thrust |

${J}_{2}$ | ${J}_{2}$ perturbation |

P | Proposed method |

I | Indirect method |

N | N-th orbital revolution |

0 | Initial time |

f | Final time |

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**Figure 4.**Comparison of $\Delta V$, for example 1. (

**a**) $\Delta V$ solved by the two methods. (

**b**) Percentage error of the estimation of $\Delta V$.

**Figure 5.**Comparison of $\Delta V$ for example 2. (

**a**) $\Delta V$ solved by the two methods. (

**b**) Percentage error of the estimation of $\Delta V$.

Case | ${\mathit{a}}_{0}$ | ${\mathit{e}}_{0}$ | ${\mathit{i}}_{0}$, deg | ${\mathit{\Omega}}_{\mathbf{0}}$, deg | ${\mathit{\omega}}_{0}$, deg | ${\mathit{f}}_{0}$, deg |
---|---|---|---|---|---|---|

1 | 1.1759, ${R}_{E}$ | 0.001 | 10 | 30 | 10 | 0 |

2 | 3.9196, ${R}_{E}$ | 0.5 | 55 | 150 | 130 | 0 |

3 | 5.8011, ${R}_{E}$ | 0.3 | 100 | 270 | 250 | 0 |

4 | 1.0, AU | 0.0167 | 5 | 30 | 50 | 0 |

Case 3 | ${\mathit{a}}_{\mathit{f}}$, ${\mathit{R}}_{\mathit{E}}$ | ${\mathit{e}}_{\mathit{f}}$ | ${\mathit{i}}_{\mathit{f}}$, deg | ${\mathit{\Omega}}_{\mathit{f}}$, deg | ${\mathit{\omega}}_{\mathit{f}}$, deg |
---|---|---|---|---|---|

Indirect method | 7.6628 | 0.4875 | 104.9775 | 275.8517 | 291.4525 |

GPOPS | 7.6775 | 0.4837 | 105.0753 | 275.8700 | 290.1070 |

Percentage error | 1.9 × 10^{−3} | 7.8 × 10^{−3} | 9.3 × 10^{−4} | 6.6 × 10^{−5} | 4.6 × 10^{−3} |

Proposed Method | Indirect Method | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Case | ${\mathit{t}}_{\mathit{a}},\times {\mathit{10}}^{-\mathit{5}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{s}$ | ${\mathit{t}}_{\mathit{e}},\times {\mathit{10}}^{-\mathit{5}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{s}$ | ${\mathit{t}}_{\mathit{i}},\times {\mathit{10}}^{-\mathit{5}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{s}$ | ${\mathit{t}}_{\mathit{\Omega}},\times {\mathit{10}}^{-\mathit{5}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{s}$ | ${\mathit{t}}_{\mathit{\omega}},\times {\mathit{10}}^{-\mathit{5}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{s}$ | ${\mathit{t}}_{\mathit{a}},\mathbf{s}$ | ${\mathit{t}}_{\mathit{e}},\mathbf{s}$ | ${\mathit{t}}_{\mathit{i}},\mathbf{s}$ | ${\mathit{t}}_{\mathit{\Omega}},\mathbf{s}$ | ${\mathit{t}}_{\mathit{\omega}},\mathbf{s}$ |

1 | 6.2 | 7.3 | S1: 0.08 | S1: 0.04 | / | 25.64 | 4.58 | 150.73 | 6.91 | / |

S2: 20 | S2: 19 | |||||||||

2 | 1.1 | 1.2 | S1: 0.05 | S1: 0.06 | 10 | 10.19 | 71.48 | 1.51 | 5.84 | 16.07 |

S2: 3.3 | S2: 3.2 | |||||||||

3 | 1.0 | 0.7 | S1: 0.09 | S1: 0.08 | 20 | 1.44 | 15.65 | 1.15 | 9.02 | 27.67 |

S2: 1.9 | S2: 1.9 | |||||||||

4 | 0.093 | 0.016 | 0.011 | 0.083 | / | 0.036 | 0.032 | 0.082 | 0.367 | / |

${\mathit{x}}_{\mathit{f}}$ | Proposed | Indirect Method | Percentage Error | |
---|---|---|---|---|

Case 1 | ${a}_{f}$, ${R}_{E}$ | 1.3287 | 1.3297 | 8.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

${e}_{f}$ | 0.0923 | 0.0928 | 6.3 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${i}_{f}$, deg | S1: 12.1615 | 12.1647 | 2.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

S2: 12.1615 | 2.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |||

${\Omega}_{f}$, deg | S1: 42.4473 | 42.6318 | 4.3 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

S2: 42.4287 | 4.7 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |||

Case 2 | ${a}_{f}$, ${R}_{E}$ | 4.8558 | 4.8658 | 2.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

${e}_{f}$ | 0.6369 | 0.6376 | 1.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${i}_{f}$, deg | S1: 59.9658 | 60.1101 | 2.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

S2: 59.9955 | 1.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |||

${\Omega}_{f}$, deg | S1: 156.6328 | 156.9221 | 1.8 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

S2: 156.8552 | 1.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |||

${\Omega}_{f}$(${J}_{2}$), deg | 152.2657 | 152.8683 | 3.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${\omega}_{f}$, deg | 147.5200 | 147.9650 | 3.0 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${\omega}_{f}$(${J}_{2}$), deg | 147.5433 | 147.9809 | 2.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

Case 3 | ${a}_{f}$, ${R}_{E}$ | 7.6366 | 7.6628 | 3.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

${e}_{f}$ | 0.4863 | 0.4875 | 2.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

${i}_{f}$, deg | S1: 104.9113 | 104.9775 | 6.3 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

S2: 104.9107 | 6.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |||

${\Omega}_{f}$, deg | S1: 275.6966 | 275.8517 | 5.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

S2: 275.6077 | 8.8 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |||

${\Omega}_{f}$(${J}_{2}$), deg | 275.8296 | 276.0224 | 6.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

${\omega}_{f}$, deg | 287.4059 | 291.4525 | 13 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${\omega}_{f}$(${J}_{2}$), deg | 287.3791 | 290.9265 | 12 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

Case 4 | ${a}_{f}$, AU | 1.2387 | 1.2684 | 2.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

${e}_{f}$ | 0.1952 | 0.2081 | 6.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

${i}_{f}$, deg | 9.2333 | 9.5141 | 2.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${\Omega}_{f}$, deg | 78.5656 | 84.9038 | 7.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

Example 1 | Example 2 | |||
---|---|---|---|---|

Orbit Elements | Initial Orbit | Target | Initial Orbit | Target |

Semi-major axis, km | 27,906 | 27,906 | 27,906 | 30,906 |

Eccentricity | $0.0106$ | $0.0106$ | 0.0106–0.4106 | free |

Inclination, deg | 40 | 42–56 | 40 | 45 |

Average Computational Time | Example 1 | Example 2 |
---|---|---|

Proposed method, $\mathrm{s}$ | $4.3\times {10}^{-4}$ | $6.1\times {10}^{-5}$ |

Indirect method, $\mathrm{s}$ | 82 | $8.8$ |

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**MDPI and ACS Style**

Wang, Z.; Cheng, L.; Jiang, F.
Approximations for Secular Variation Maxima of Classical Orbital Elements under Low Thrust. *Mathematics* **2023**, *11*, 744.
https://doi.org/10.3390/math11030744

**AMA Style**

Wang Z, Cheng L, Jiang F.
Approximations for Secular Variation Maxima of Classical Orbital Elements under Low Thrust. *Mathematics*. 2023; 11(3):744.
https://doi.org/10.3390/math11030744

**Chicago/Turabian Style**

Wang, Zhaowei, Lin Cheng, and Fanghua Jiang.
2023. "Approximations for Secular Variation Maxima of Classical Orbital Elements under Low Thrust" *Mathematics* 11, no. 3: 744.
https://doi.org/10.3390/math11030744