<aside> ⏩ ChatGPT wrote all of this! Gravity Gradient Torques is the only section that has be manually verified so far.

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External Torques on a Satellite

External torques are forces acting on a satellite that cause it to rotate or change its orientation in space. These torques arise from various environmental interactions and can significantly affect the satellite's stability and control. The primary external torques include Gravity Gradient Torque, Aerodynamic Drag Torque, Solar Radiation Pressure Torque, and Magnetic Torque.

1. Gravity Gradient Torques

Explanation: Gravity gradient torque arises due to the variation in gravitational force across the satellite's structure. This effect is significant for satellites in low Earth orbit (LEO) where the difference in gravitational pull between different parts of the satellite creates a torque that tends to align the satellite's long axis with the local vertical direction.

Equation: The gravity gradient torque ($\mathbf{T}{gg}$) can be calculated using: $\mathbf{T}{gg} = 3 \frac{\mu}{r^3} \left[ \mathbf{r} \times (\mathbf{I} \mathbf{r}) \right]$ where:

• $\mu$ is the Earth's gravitational constant.
• $r$ is the distance from the Earth's center to the satellite.
• $\mathbf{r}$ is the position vector of the satellite relative to the Earth's center.
• $\mathbf{I}$ is the satellite's moment of inertia tensor.

Gravity Gradient

2. Aerodynamic Drag Torque

Explanation: Aerodynamic drag torque occurs when a satellite passes through the Earth's atmosphere. The interaction between the atmospheric particles and the satellite's surface generates a force that can create a torque, especially if the force is not evenly distributed over the satellite's surface.

Equation: The aerodynamic drag torque ($\mathbf{T}{drag}$) can be expressed as: $\mathbf{T}{drag} = \frac{1}{2} \rho v^2 C_d A (\mathbf{r}{cp} - \mathbf{r}{cm})$ where:

• $\rho$ is the atmospheric density.
• $v$ is the relative velocity of the satellite to the atmosphere.
• $C_d$ is the drag coefficient.
• $A$ is the reference area.