Magnetic Field Centered Dipole Planet Model

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Description

The Magnetic Field Centered Dipole model is a generic magnetic field model that can be parameterized to be centered about any object with an aligned vector. The centered dipole model is a first-order result of the more complex spherical harmonic modelling of the planet’s magnetic field. The model can be used to provide magnetic field estimates of the spacecraft orbiting the Earth or any other body configured with a magnetic field. By adjusting the parameters, the magnetic field can be added to another planet.

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Example Use Cases

• Simulate magnetorquers for desaturation of reaction wheels.
• Determine the magnetic field vector at unique locations in the orbit.
• Using a magnetometer for attitude control across a low-Earth orbit.

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Module Implementation

The spacecraft’s location relative to the planet frame is required. Let $r_{B/P}$ be the spacecraft position vector relative to the planet centre. In the simulation, the spacecraft location is given relative to an inertial frame origin $\ast \ast O\ast \ast$. The planet-centric position vector is computed using

$${\mathbf{r}}_{B/P}={\mathbf{r}}_{B/O}-{\mathbf{r}}_{P/O}$$

If no planet ephemeris message is specified, then the planet position vector $r_{P/O}$ is set to zero. Let [$PN$] be the direction cosine matrix that relates the rotating planet-fixed frame relative to an inertial frame $N:\{{\hat{n}}_1,{\hat{n}}_2,{\hat{n}}_3\}$. The simulation provides the spacecraft position vector in inertial frame components. The planet-centric position vector is then written in Earth-fixed frame components using

$${}^P{\mathbf{r}}_{B/P}=[PN{]}^N{\mathbf{r}}_{B/P}$$

The centrered dipole model is a first-order result of the more complex spherical harmonic modelling of the planet’s magnetic field. There are several solutions that provide an answer in the local North-Earth-Down or NED frame, or in the local spherical coordinates. Let m be the magnetic dipole vector which is then defined as

$$V({\mathbf{r}}_{B/P})=\frac{\mathbf{m}\cdot {\mathbf{r}}_{B/P}}{|{\mathbf{r}}_{B/P}{|}^3}$$

with the dipole vector being defined in Earth fixed frame $E$ coordinates as

$$\mathbf{m}{=}^E\left[\begin{array}{c}{g}_1^1\\ b_1^1\\ c_1^0\end{array}\right]$$

The magnetic field vector B is expressed at the spacecraft location as

$$B(r_{B/P})=-\nabla V(r_{B/P})=\frac{3(\mathbf{m}\cdot {\mathbf{r}}_{B/P}){\mathbf{r}}_{B/P}-{\mathbf{r}}_{B/P}\cdot {\mathbf{r}}_{B/P}\mathbf{m}}{|{\mathbf{r}}_{B/P}{|}^5}=\frac{1}{|r_{B/P}{|}^3}(3(\mathbf{m}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m})$$

where

$$\hat{\mathbf{r}}=\frac{{\mathbf{r}}_{B/P}}{|{\mathbf{r}}_{B/P}|}$$

The above vector equation is evaluated in E-frame components, while the output is mapped into $N$-frame components by returning

$${}^N\mathbf{B}=[PN{]}^T{}^P\mathbf{B}$$

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Assumptions / Limitations

• The field, unless specified, will assume to exist everywhere in $N$ space with some value
• All spacecraft within the simulation will be affected by all fields that have been added.

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References

[1] Hanspeter Schaub and John L. Junkins. Analytical Mechanics of Space Systems. AIAA Education Series, Reston, VA, 4th edition, 2018. [2] F. Landis Markley and John L. Crassidis. Fundamentals of Spacecraft Attitude Determination and Control. Springer, New York, 2014. [3] Michael D. Grifin and James R. French. Space Vehicle Design. AIAA Education Series, Reston, VA, 2005.