How to (III): Calculate Satellite Disturbance Torques

Mini Series: Designing a Satellite for Dummies

Are you an aspiring aerospace engineer, a space enthusiast, a parent checking your child’s homework or simply interested in the specifics of how to design certain satellite parts? Then this is the place to be.

In this mini series we will go through the basics of designing and scaling a satellite, ranging from solar arrays to propellant tanks and even orbital parameters. If you would like us to cover other space-related topics, feel free to reach out to engineering@valispace.com.

Part 3: How to Calculate Satellite Disturbance Torques

In the first two tutorials of this series we discussed components of the satellite’s power system. We will now explain the necessary steps for designing the Attitude Determination and Control System (ADCS). During the lifetime of an Earth-bound satellite, its attitude (the direction it is pointed towards) gets continually disturbed by different forces, called disturbance torques. These torques have to be mitigated by the ADCS, to make sure the satellite is always pointed in the right direction. In this tutorial, we will discuss how to calculate these disturbance torques.

Different types of torques

We distinguish four different types of disturbance torques, namely gravity gradient, solar radiation pressure, magnetic field and aerodynamic torques. Because the ADCS must handle the maximum possible torques, we will try to find the worst-case torques for all different cases. Note that some of these forces can also cause orbit perturbations, but in this tutorial we will just focus on the attitude of the satellite. Below you can see the influence of the different torques on a typical satellite over a range of altitudes. It shows that generally, up to an altitude of around 500 km, the aerodynamic torque is the maximum torque, but going higher up the gravity gradient will take over.

The Earth’s gravitational field varies inversely with distance from the Earth, thus the gravitational field varies slightly across a satellites length. This difference in acceleration causes a torque on the satellite. As a first order approximation, this torque is constant along the satellite’s orbit for an Earth-orientated vehicle and can be approximated as follows:

$$T_g=\frac{3\mu }{2R^3}|I_z–I_y|sin\left(2\theta \right)\left[Nm\right]$$

In here, $\mu$ is Earth’s gravitational constant ($398600k{m}^3/s^2$) and R is the orbital radius (dependent on the satellite’s altitude). $I_z$ and $I_y$ are the moments of inertia referring to the z- and y-axis. The moments of inertia are highly variable depending on the exact satellite design, to find out how to calculate them, give these different sources a look. Note that $I_y$ is interchangeable with $I_z$, and as we are trying to find the worst-case torque, the moment of inertia which gives the highest torque should be used. $\Theta$ is the maximum angle the local vertical makes from the z-axis, as visualized below.

Solar radiation pressure torque

Radiation (electromagnetic waves) carries momentum according to Maxwell’s theory of electromagnetism. Upon hitting a surface, this momentum can be transferred to the surface. The amount of momentum transferred depends on the type of surface being illuminated. It will be lowest for transparent surfaces, higher for absorbent surfaces and highest for reflective surfaces. In general, it can be said that solar arrays are absorbers and a spacecraft’s body is a reflector. The total force exerted on a spacecraft by solar radiation is as follows:

$$F=\frac{J_s}{c}{A}_{s}cos\left(I\right)(1+q)\left[N\right]$$

Here, $J_s=1367W/m^2$ is the solar constant at Earth, $c=3\cdot {10}^{8}m/s$ is the speed of light, $A_s$ is the total surface area and $I$ is the angle of incidence of the solar radiation. Finally, $q$ is the reflectance factor. Obviously, this value is not constant over the spacecraft’s body, but with a first order approximation it can be assumed to be constant and equal to 0.6. Now, the torque exerted on the satellite will be:

$$T_{sp}=F(c_{ps}–cg)\left[Nm\right]$$

Here, $cg$ is the center of gravity and $c_{ps}$ is the center of solar pressure. This center can be found in a similar way to finding the center of gravity, namely by finding the surfaces which are hit by radiation and their areas as follows:

$$c_{ps}=\frac{\Sigma A_n{x}_n}{\Sigma A_n}\left[m\right]$$

Magnetic field torque

The magnetic field of the Earth causes a cyclic torque across the spacecraft no matter its orientation due to interactions with the spacecraft’s magnetic dipole. It can be approximated using the following formula:

$$T_m=DB=\frac{{\mu }_E}{c{R}^3}\left[Nm\right]$$

In here, $B$ is the Earth’s magnetic field strength, which is approximated using the magnetic moment of Earth  ${\mu }_E=7.96\cdot {10}^{15}tesla\cdot m^3$, c is an approximation constant, which can be taken to be 1 for a polar orbit and as 2 for an equatorial orbit, and R is the distance to Earth’s dipole center in meters. $D$ is the satellite’s residual dipole, which is caused by electric currents and magnetic material within the vehicle. In the design of your satellite, it is important to minimize this value, for instance by including twist in the cables and avoiding large loops on the electronics boards! Due to different currents over time, this is hard to calculate. However, $D$ can be estimated as a function of mass as:

$$D=c\cdot {10}^{-3}\cdot m_{sc}\left[A{m}^2\right]$$

Here, $m_{sc}$ is the satellite mass and c is a constant in a range of 1 to 10 depending on the level of magneticity of the spacecraft (NASA source). Also, the residual dipole can be measured with a magnetic moment test after the satellite is integrated!

Aerodynamic torque

An object flying through air is subjected to aerodynamic drag. For some low-flying satellites, this means that it is necessary to calculate torques generated aerodynamically across the vehicle. For satellites above an altitude of $500km$ this can generally be neglected. This can be done using the following approximation:

$$T_a=0.5\left[\rho C_{d}A{V}^2\right](c_{pa}–cg)\left[Nm\right]$$

In here, $\rho$ is the atmospheric density, $C_d$ is the drag coefficient (usually between 2 and 2.5), $A$ is the area of the front-facing satellite, $V$ is the spacecraft’s velocity and $cg$ is the center of gravity. $c_{pa}$ is the center of aerodynamic pressure, which can be calculated in a similar way as $c_{ps}$ is calculated.

If you followed the steps correctly, you have now calculated the different disturbance torques on your satellite and you are ready to start the sizing of your ADCS, congratulations!

We hope you liked this mini-tutorial! If you want to learn how the disturbance torques and the ADCS are related to other subsystems in a satellite or how to design a complete satellite using Valispace and practical examples, also check our Satellite Tutorial.
Stay tuned for more and feel free to give us feedback at contact-us@valispace.com!