# Controlling the Perturbations of Solar Radiation Pressure on the Lorentz Spacecraft

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

^{2}for an atmospherical model with an exospheric temperature of 1400 k [1].

## 2. Perturbations in Orbital Elements

- a is the semi–major axis;
- e is the eccentricity;
- I is the inclination;
- $\mathsf{\Omega}$ is the longitude of ascending node;
- $\tilde{\omega}$ is the longitude of the perigee;
- $\u03f5$ is the mean longitude at epoch.

## 3. Orbital Averaging

## 4. Effect of Solar Radiation Pressure

^{2}, so the RP is about $4.6$ MPa (absorbed) [28].

- $\mathcal{C}$ is the light speed;
- ${r}_{0}$ is the distance between the Earth and Sun;
- ${r}_{1}$ is the distance between the satellite and Sun;
- $\alpha $ is represents the reflection coefficient of the surface;
- $\gamma $ is an incident angle of falls ray at to the surface;
- ${s}_{0}$ is a solar constant due to the mean distance between the Earth and Sun;
- $\mathbf{S}$ is a unit vector in the direction of Satellite–Sun given in a geocentric equatorial frame;
- $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are unit vectors in the geocentric coordinates system;
- $\theta $ is the true longitude of the Sun;
- $\epsilon $ is the obliquity of ecliptic.

#### 4.1. Variations of the Longitude of Ascending Node Due to Solar Radiation Pressure

#### 4.2. Variations of the Argument of Perigee Due to Solar Radiation Pressure

## 5. Effect of Lorentz Force

- B is the moment of magnetic dipole of planet;
- q is the charge per unit mass of satellite;
- $\upsilon $ is the planetary rotation speed.

#### 5.1. Variations of the Longitude of Ascending Node Due to Lorentz Force

#### 5.2. Variations of the Argument of Perigee Due to Lorentz Force

## 6. Balancing of Solar Radiation Pressure Effect

#### 6.1. Controlling on the Longitude of Ascending Node

#### 6.2. Controlling on the Argument of Perigee

## 7. Numerical Results

^{3}, $\mu =\mathrm{398,600}$ km

^{3}/s

^{2}, $\upsilon =\pi /12$ h

^{−1}, $\epsilon =23.439\xb0$, and $i=\pi /4$. The calculations have been estimated when the semi-major axes ($a=7100$ km) and the constant ${\mathcal{R}}_{\mathcal{C}}=9.2\times {10}^{-5}$ for Low Earth Orbit while the same quantities are ($a=\mathrm{42,160}$ km) and ${\mathcal{R}}_{\mathcal{C}}=4.2\times {10}^{-5}$ for Geosynchronous Orbit along with the true longitude of the Sun ($\theta =0$) in both two cases.

#### 7.1. Results of Longitude of Ascending Node

#### 7.1.1. Evolution of Charge Quantity Versus Argument of Perigee

#### 7.1.2. Evolution of Charge Quantity Versus Longitude of Ascending Node

#### 7.1.3. Evolution of Charge Quantity Versus Eccentricity

#### 7.2. Results of Argument of Perigee

#### 7.2.1. Evolution of Charge Quantity Versus Argument of Perigee

#### 7.2.2. Evolution of Charge Quantity Versus Longitude of Ascending Node

#### 7.2.3. Evolution of Charge Quantity versus Eccentricity

#### 7.3. Implications on Spacecraft Attitude

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Charge value versus the argument of perigee with different values for the longitude of node at Low Earth Orbit (LEO).

**Figure 2.**Charge value versus the argument of perigee with different values for the eccentricity at LEO.

**Figure 3.**Charge value versus the argument of perigee with different values for the longitude of node at Geosynchronous Orbit (GSO).

**Figure 4.**Charge value versus the argument of perigee with different values for the eccentricity at GSO.

**Figure 5.**Charge value versus the longitude of node with different values for the argument of perigee at LEO.

**Figure 6.**Charge value versus the longitude of node with different values for the eccentricity at LEO.

**Figure 7.**Charge value versus the longitude of node with different values for the argument of perigee at GSO.

**Figure 8.**Charge value versus the longitude of node with different values for the eccentricity at GSO.

**Figure 9.**Charge value versus the eccentricity with different values for the longitude of node at LEO.

**Figure 10.**Charge value versus the eccentricity with different values for the longitude of node at GSO.

**Figure 11.**Charge value versus the argument of perigee at different values for the longitude of node at LEO.

**Figure 12.**Charge value versus the argument of perigee at different values for the eccentricity at LEO.

**Figure 13.**Charge value versus the longitude of node at different values for the argument of perigee at LEO.

**Figure 14.**Charge value versus the longitude of node at different values for the eccentricity at LEO.

**Figure 15.**Charge value versus the eccentricity at different values for the longitude of node at LEO.

**Figure 16.**Charge value versus the eccentricity at different values for the argument of perigee at LEO.

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**MDPI and ACS Style**

Mostafa, A.; El-Saftawy, M.I.; Abouelmagd, E.I.; López, M.A.
Controlling the Perturbations of Solar Radiation Pressure on the Lorentz Spacecraft. *Symmetry* **2020**, *12*, 1423.
https://doi.org/10.3390/sym12091423

**AMA Style**

Mostafa A, El-Saftawy MI, Abouelmagd EI, López MA.
Controlling the Perturbations of Solar Radiation Pressure on the Lorentz Spacecraft. *Symmetry*. 2020; 12(9):1423.
https://doi.org/10.3390/sym12091423

**Chicago/Turabian Style**

Mostafa, A., M. I. El-Saftawy, Elbaz I. Abouelmagd, and Miguel A. López.
2020. "Controlling the Perturbations of Solar Radiation Pressure on the Lorentz Spacecraft" *Symmetry* 12, no. 9: 1423.
https://doi.org/10.3390/sym12091423