Passive magnetic detumbling to enable Active Debris Removal of non-operational satellites in Low Earth Orbit

Abstract

Under the umbrella of the Clean Space Initiative, ESA has promoted activities in the area of Active Debris Removal (ADR). One of the main challenges driving the complexity of the rendezvous and capture of debris is its tumbling motion. It is highly desirable that kinetic energy dissipation devices damp the angular rates of debris in advance. Unfortunately, Eddy currents in conductive structures only provide a very slow detumbling. A potentially faster solution proposed by ESA (patent pending Ref EP19182205) consists in short-circuiting the satellite Magnetic Torquers already on board LEO satellites. The tumbling motion of the satellite within the Earth magnetic field results in the induction of currents in the Magnetic Torquers coils, dissipating energy by Joule heating. With a spin rate around 1 rpm, the Gravity Gradient torques create a slow precession of the satellite angular momentum around the orbit normal while the magnetic torques generated by a potential satellite residual dipole command a perpendicular precession around the North Pole. In parallel, the short-circuited Magnetic Torquers create damping and tilting magnetic torques. The spin axis re-orientation and the reduction of the angular rates are analytically predicted. A reduced set of driving parameters (debris inertias, altitude, spin axis orientation, and Magnetic Torquers characteristics) are identified that estimate the detumbling durations. ESA aims to further study and implement this concept with industry during the procurement phase of the new Copernicus Expansion Sentinels kicked off in autumn 2020, with first launches in 2025. The increased definition of the six satellites will permit to consider the positive contribution of Eddy currents in the metallic structures as well as the detrimental impact of the Solar Radiation pressure on tumbling Solar Arrays.

Appendices

Appendix A

Simulator architecture

A Python simulator was used to simulate the magnetic detumbling of the spacecraft. With this simulator, it is possible to run a simulation of 100 days in 30 min with a time step of 0.1 s. This performance is achieved using the Numba library to speed up the code and use an LSODA ODE solver.

Figure 30 shows the architecture of the simulator, it is organized around three differential equations that will be described below. The simulator includes gravitational perturbation for the orbit propagation and three different type of torque for the attitude.

Fig. 30

Simulator architecture

The Kinematic and Dynamic model is based on three differential equations:

Orbital propagation:

$$\ddot{r}=-\frac{\mu }{{\Vert r\Vert }^{3}} r+Perturbation$$

(137)

The orbit is propagated in the inertial frame meaning that the perturbations also need to be calculated in the inertial frame.

Euler equation for Rigid Body Kinematics: (no flexible modes)

$$I\dot{{w}_{B}}=-{w}_{B}\times I{w}_{B}+{T}_{b}$$

(138)

In this equation, the spacecraft angular velocity is propagated using current velocity and the torques applied on the spacecraft. These torques need to be calculated.

Quaternion/Attitude dynamics equation: (with $q={q}_{1}i+{q}_{2}j+{q}_{3}k+{q}_{4}$)

$${\dot{q}}_{\mathrm{1,2},3}=\frac{1}{2}\left[{q}_{4}{w}_{b}-{w}_{b}\times {q}_{\mathrm{1,2},3}\right] and \dot{{q}_{4}}=-\frac{1}{2}\left[{w}_{{b}_{x}}{q}_{1}+{w}_{{b}_{y}}{q}_{2}+{{{w}_{ }}_{b}}_{z}{q}_{3}\right]$$

(139)

This equation allows computing the spacecraft attitude in the body frame. With this equation, it is important to normalize the calculated quaternion at each step to avoid numerical issues over time. This quaternion is also used to calculate rotational matrix to convert data from inertial to body frame and vice versa.

The J2–J6 gravitational disturbances are implemented and injected into the orbit propagator equation. They are computed one by one using the coefficients in Table

Table 10 J2–J6 Spherical harmonics coefficients

10 and the spherical harmonic geopotential model from [20].

The Earth Magnetic Field is computed using the IGRF-13 Model, this is a significant improvement with respect to the analytical framework which assumes a simple dipole model, no magnetic declination and a polar orbit (ie Earth magnetic field lines in the orbital plane) (Fig. 

Fig. 31

Simulator IGRF-13 model magnetic field

31).

The external torques represented in this simulator are the ones which have a predominant effect. Aerodynamic torques and solar torques are not modelled.

Gravity Gradient Torque:

$${T}_{GG}=\frac{3\mu }{{\Vert {r}_{B}\Vert }^{5}}\left[\begin{array}{c}{{r}_{B}}_{y}{{r}_{B}}_{z}({I}_{z}-{I}_{y})\\ {{r}_{B}}_{x}{{r}_{B}}_{z}({I}_{x}-{I}_{z})\\ {{r}_{B}}_{x}{{r}_{B}}_{y}({I}_{y}-{I}_{x})\end{array}\right]$$

(140)

The calculation of the gravity gradient torque in the body frame is done using the spacecraft Inertia and the orbital position also in body frame.

Magnetic Torque created by the Satellite Residual Dipole:

$${\overrightarrow{T}}_{RD}={\overrightarrow{\mathcal{M}}}_{RD}\times {\overrightarrow{B}}_{Earth}$$

(141)

This equation describes the residual magnetic dipole torque generated by the metallic part of the spacecraft in the Earth magnetic field. ${\overrightarrow{\mathcal{M}}}_{RD}$ is the magnetic moment of the spacecraft and ${\overrightarrow{B}}_{Earth}$ is the Earth magnetic field in the body frame.

Magnetic Torque created by the short-circuited magnetic torquers:

$${\overrightarrow{T}}_{SC}= -\left[\mathrm{M}{\left(\frac{{\overrightarrow{dB}}_{Earth}}{dt}\right)}_{sat}\right] \times {\overrightarrow{B}}_{Earth }= \overrightarrow{{T}_{SC1}}+ \overrightarrow{{T}_{SC2}}$$

(142)

This equation is the key equation for the magnetic damping, the torque is generated by the local change of magnetic induction due to the spacecraft rotation ($\overrightarrow{{T}_{SC1}}$) and by the variation due to the displacement of the spacecraft along its orbit ($\overrightarrow{{T}_{SC2}}$).

This macroscopic model is taken from the derivation of the Magnetic Tensor presented in previous sections. The added value of the simulator is that it computes instantaneous torques and their immediate dynamic effect. The analytical framework assumes that the angular momentum remains quasi inertial over one orbit, computes the averaged torques (Gravity Gradient, Residual dipole and short-circuited magnetic torquers) and their effects on the angular momentum evolution from one orbit to the next one. The time step of the simulator must be sufficiently small (typ. 0.1 s) to avoid numerical artefacts, like for instance an artificial damping of the nutation event when no short-circuited magnetic torquers are dissipating energy.

Appendix B

Gravity Gradient torque acting on a spinner

This appendix develops results which can be found in textbooks, in particular [19], and adds the case of a precessing orbit.

4.1 Average Gravity Gradient torque over one spin period

It first details the derivation of the mean Gravity Gradient torque applied on a spinning satellite used in Sect. 2.1.1, starting from Eq. (6) reminded hereunder:

$${\langle \overrightarrow{{T}_{GG}}\rangle }_{spin}= \frac{1}{{T}_{spin}} {\int }_{0}^{{T}_{spin}}\overrightarrow{{T}_{GG}}(t) dt$$

(143)

The inertial frame ${\mathcal{R}}_{1}$ (X₁, Y₁, Z₁) with Z₁ = Z_p allows to have constant vectors in the integration (Fig. 32). The transformation matrix ${\mathcal{R}}_{1}$ (X₁, Y₁, Z₁) ${\mathcal{R}}_{{\varvec{p}}}$ (X_p, Y_p, Z_p) is simply:

$$M= \left(\begin{array}{ccc}\mathrm{cos}{\omega }_{z}t& -\mathrm{sin}{\omega }_{z}t& 0\\ \mathrm{sin}{\omega }_{z}t& \mathrm{cos}{\omega }_{z}t& 0\\ 0& 0& 1\end{array}\right)$$

(144)

$$ \begin{gathered} \left\langle {T_{GG} } \right\rangle_{{spin, R_{1} }} = \frac{{3\omega_{0}^{2} }}{{T_{spin} }} R_{{sR_{1} }} \times \mathop \smallint \limits_{0}^{{T_{spin} }} M \mathop I\limits^{ = } M^{ - 1} dt{\kern 1pt} .{\kern 1pt} R_{{sR_{1} }} \hfill \\ \hfill \\ \end{gathered} $$

(145)

Fig. 32

Inertial reference frame ${\mathcal{R}}_{1}$

since $\overrightarrow{{R}_{s}}$ is assumed to remain inertial over a spin period.

$$ M \mathop I\limits^{ = } M^{ - 1} = \left( {\begin{array}{*{20}c} {I_{x} \cos^{2} \omega_{z} t + I_{y} \sin^{2} \omega_{z} t } & {\left( {I_{y} - I_{x} } \right) \sin \omega_{z} t\cos \omega_{z} t} & 0 \\ {\left( {I_{y} - I_{x} } \right) \sin \omega_{z} t\cos \omega_{z} t} & {I_{x} \sin^{2} \omega_{z} t + I_{y} \cos^{2} \omega_{z} t} & 0 \\ 0 & 0 & {I_{z} } \\ \end{array} } \right)_{{R_{1} }} $$

(146)

$$ \frac{1}{{T_{spin} }}\mathop \smallint \limits_{0}^{{T_{spin} }} M \mathop I\limits^{ = } M^{ - 1} dt = \left( {\begin{array}{*{20}c} {\frac{{I_{x} + I_{y} }}{2}} & 0 & 0 \\ 0 & {\frac{{I_{x} + I_{y} }}{2}} & 0 \\ 0 & 0 & {I_{z} } \\ \end{array} } \right)_{{R_{1} }} $$

(147)

$${\langle {T}_{GG}\rangle }_{spin,{\mathcal{R}}_{1}}=3{\omega }_{0}^{2}{{R}_{s}}_{{\mathcal{R}}_{1}}\times {\left(\begin{array}{ccc}\frac{{I}_{x}+{I}_{y}}{2}& 0& 0\\ 0& \frac{{I}_{x}+{I}_{y}}{2}& 0\\ 0& 0& {I}_{z}\end{array}\right)}_{{\mathcal{R}}_{1}} {{R}_{s}}_{{\mathcal{R}}_{1}}=3{\omega }_{0}^{2}{\left(\begin{array}{c}{R}_{x}^{1}\\ {R}_{y}^{1}\\ {R}_{z}^{1}\end{array}\right)}_{{\mathcal{R}}_{1}} \times {\left(\begin{array}{c}{ \frac{{I}_{x}+{I}_{y}}{2}R}_{x}^{1}\\ \frac{{I}_{x}+{I}_{y}}{2} {R}_{y}^{1}\\ {I}_{z} {R}_{z}^{1}\end{array}\right)}_{{\mathcal{R}}_{1}}$$

(148)

$${\langle {{\varvec{T}}}_{{\varvec{G}}{\varvec{G}}}\rangle }_{spin,{\mathcal{R}}_{1}}=3{\omega }_{0}^{2} \left({I}_{z}-\frac{{I}_{x}+{I}_{y}}{2}\right){R}_{z}^{1}{\left(\begin{array}{c}{R}_{y}^{1}\\ {-R}_{x}^{1}\\ 0\end{array}\right)}_{{\mathcal{R}}_{1}}$$

(149)

In vector notation, this spin-average torque can simply be written:

$${\langle \overrightarrow{{{\varvec{T}}}_{{\varvec{G}}{\varvec{G}}}}\rangle }_{{\varvec{s}}{\varvec{p}}{\varvec{i}}{\varvec{n}}}=\boldsymbol{ }3{{\varvec{\omega}}}_{0}^{2}\boldsymbol{ }\left[{{\varvec{I}}}_{{\varvec{z}}}-\left(\frac{{{\varvec{I}}}_{{\varvec{x}}}+{{\varvec{I}}}_{{\varvec{y}}}}{2}\right)\right]\left(\overrightarrow{{{\varvec{R}}}_{{\varvec{s}}}}.\overrightarrow{{{\varvec{Z}}}_{{\varvec{p}}}}\right)\left(\overrightarrow{{{\varvec{R}}}_{{\varvec{s}}}}\times \overrightarrow{{{\varvec{Z}}}_{{\varvec{p}}}}\right)$$

(150)

4.2 Average Gravity Gradient torque over one polar orbit

In order to have constant vectors in the integration, the Earth vector R_s needs now to be expressed in the inertial frame ${\mathcal{R}}_{0}$ (X₀, Y₀, Z₀), with Z₀ being the orbit normal, as shown on Fig. 1. The principal axis Z_p keeps constant components in this inertial frame ${\mathcal{R}}_{0}$. The average torque over one orbit can be derived by a proper integration of Eq. (8) reminded hereunder:

$${\langle {T}_{GG}\rangle }_{orbit}= \frac{1}{{T}_{orbit}} {\int }_{0}^{{T}_{orbit}}3{\omega }_{0}^{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]\left(\overrightarrow{{R}_{s}}.\overrightarrow{{Z}_{p}}\right)\left(\overrightarrow{{R}_{s}}\times \overrightarrow{{Z}_{p}}\right) dt$$

(151)

We will express the Earth vector R_s in the inertial frame ${\mathcal{R}}_{0}$ (X₀, Y₀, Z₀) in order to have constant vectors in the integration with Z₀ being the orbit normal (Fig. 33). We will assume that the principal axis Z_p keeps constant components in this inertial frame ${\left(\begin{array}{c}{Z}_{x}^{0}\\ {Z}_{y}^{0}\\ {Z}_{z}^{0}\end{array}\right)}_{{\mathcal{R}}_{0}}$

$$\overrightarrow{{R}_{s}}.\overrightarrow{{Z}_{p}}= \mathrm{cos}{\omega }_{0}t {Z}_{x}^{0}+ \mathrm{sin}{\omega }_{0}t {Z}_{y}^{0}$$

(152)

$$\left({{R}_{s}}_{{\mathcal{R}}_{0}}\times {{Z}_{p}}_{{\mathcal{R}}_{0}}\right)= {\left(\begin{array}{c}\mathrm{sin}{\omega }_{0}t {Z}_{z}^{0} \\ - \mathrm{cos}{\omega }_{0}t {Z}_{z}^{0} \\ \mathrm{cos}{\omega }_{0}t {Z}_{y}^{0} -\mathrm{sin}{\omega }_{0}t {Z}_{x}^{0} \end{array}\right)}_{{\mathcal{R}}_{0}}$$

(153)

$${\langle {T}_{GG}\rangle }_{orbit,{\mathcal{R}}_{0}}= \frac{1}{{T}_{orbit}} {\int }_{0}^{{T}_{orbit}}3{\omega }_{0}^{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]\left(\overrightarrow{{R}_{s}}.\overrightarrow{{Z}_{p}}\right)\left(\overrightarrow{{R}_{s}}\times \overrightarrow{{Z}_{p}}\right) dt$$

(154)

$${\langle {{\varvec{T}}}_{{\varvec{G}}{\varvec{G}}}\rangle }_{orbit,{\mathcal{R}}_{0}}= 3{\omega }_{0}^{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]\frac{1}{{T}_{orbit}} {\int }_{0}^{{T}_{orbit}}{\left(\begin{array}{c}\mathrm{sin}{\omega }_{0}t\mathit{ }\mathrm{cos}{\omega }_{0}t {Z}_{x}^{0} {Z}_{z}^{0}+ {\mathrm{sin}}^{2}{\omega }_{0}t {Z}_{y}^{0} {Z}_{z}^{0}\\ {-\mathrm{cos}}^{2}{\omega }_{0}t {Z}_{x}^{0} {Z}_{z}^{0}-\mathrm{ sin}{\omega }_{0}t\mathit{ }\mathrm{cos}{\omega }_{0}t {Z}_{y}^{0} {Z}_{z}^{0}+ \\ \left({\mathrm{cos}}^{2}{\omega }_{0}t - {\mathrm{sin}}^{2}{\omega }_{0}t \right){Z}_{x}^{0} {Z}_{y}^{0}-\mathrm{ cos}{\omega }_{0}t \mathrm{sin}{\omega }_{0}t \left({{Z}_{y}^{0}}^{2}\right) \end{array}\right)}_{{\mathcal{R}}_{0}}dt$$

(155)

$$ \left\langle {\user2{T}_{{\user2{GG}}} } \right\rangle _{{orbit,\;R_{0} }} = ~3\omega _{0}^{2} ~\left[ {I_{z} - \left( {\frac{{I_{x} + I_{y} }}{2}} \right)} \right]\left( { - ~\begin{array}{*{20}c} {Z_{y}^{0} ~Z_{z}^{0} /2} \\ {Z_{x}^{0} ~Z_{z}^{0} /2} \\ 0 \\ \end{array} } \right)_{{R_{0} }} = \frac{{3\omega _{0}^{2} }}{2}~\left[ {I_{z} - \left( {\frac{{I_{x} + I_{y} }}{2}} \right)} \right]Z_{z}^{0} \left( { - ~\begin{array}{*{20}c} {Z_{y}^{0} } \\ {Z_{x}^{0} } \\ 0 \\ \end{array} } \right)_{{R_{0} }} $$

(156)

Fig. 33

Inertial reference frame R₀

In vector notation, this orbit-average torque can simply be written:

$${\langle \overrightarrow{{{\varvec{T}}}_{{\varvec{G}}{\varvec{G}}}}\rangle }_{{\varvec{o}}{\varvec{r}}{\varvec{b}}{\varvec{i}}{\varvec{t}}}=\boldsymbol{ }\frac{3{{\varvec{\omega}}}_{0}^{2}}{2}\boldsymbol{ }\left[{{\varvec{I}}}_{{\varvec{z}}}-\left(\frac{{{\varvec{I}}}_{{\varvec{x}}}+{{\varvec{I}}}_{{\varvec{y}}}}{2}\right)\right]\left(\overrightarrow{{{\varvec{Z}}}_{0}}.\overrightarrow{{{\varvec{Z}}}_{{\varvec{p}}}}\right)\left(\overrightarrow{{{\varvec{Z}}}_{{\varvec{p}}}}\times \overrightarrow{{{\varvec{Z}}}_{0}}\right)$$

(157)

4.3 Average Gravity Gradient torque over one precessing orbit

We will now consider that the orbit is a Sun Synchronous orbit, with a slight precession of the orbit $\dot{\alpha }$ (360 degrees per year) and still no significant angular momentum drift during one orbit and no nutation.

${\mathcal{R}}_{{\varvec{o}}{\varvec{r}}{\varvec{b}}}$ (X_orb, Y_orb, Z_orb) is no longer an inertial frame and is no longer identical to the inertial frame ${\mathcal{R}}_{0}$ as shown on Fig. 34.

Fig. 34

Drifting sun synchronous orbit

We get the following equations:

$$\overrightarrow{{X}_{orb}}=\mathrm{cos}\alpha \overrightarrow{{X}_{0}}-\mathrm{sin}\alpha \overrightarrow{{Z}_{0}}$$

$$\overrightarrow{{Y}_{orb}}= \overrightarrow{{Y}_{0}}$$

$$\overrightarrow{{Z}_{orb}}= \mathrm{sin}\alpha \overrightarrow{{X}_{0}}+\mathrm{cos}\alpha \overrightarrow{{Z}_{0}}$$

(158)

$$\overrightarrow{{R}_{s}}= \mathrm{cos}{\omega }_{0}t \overrightarrow{{X}_{orb}}+\mathrm{sin}{\omega }_{0}t \overrightarrow{{Y}_{orb}}$$

(159)

$$\overrightarrow{{Z}_{p}}= {\left(\begin{array}{c}{Z}_{x}^{orb}\\ {Z}_{y}^{orb}\\ {Z}_{z}^{orb}\end{array}\right)}_{{\mathcal{R}}_{orb}}$$

(160)

In the non-inertial orbital frame, we cannot integrate the components of the following equation since the reference vectors are not inertial:

$${\langle {T}_{GG}\rangle }_{orbit}= 3{\omega }_{0}^{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]\frac{1}{{T}_{orbit}} {\int }_{0}^{{T}_{orbit}}{\left(\begin{array}{c}\mathrm{sin}{\omega }_{0}t\mathit{ }\mathrm{cos}{\omega }_{0}t {Z}_{x}^{0} {Z}_{z}^{0}+ {\mathrm{sin}}^{2}{\omega }_{0}t {Z}_{y}^{0} {Z}_{z}^{0}\\ {-\mathrm{cos}}^{2}{\omega }_{0}t {Z}_{x}^{0} {Z}_{z}^{0}-\mathrm{ sin}{\omega }_{0}t\mathit{ }\mathrm{cos}{\omega }_{0}t {Z}_{y}^{0} {Z}_{z}^{0}+ \\ \left({\mathrm{cos}}^{2}{\omega }_{0}t - {\mathrm{sin}}^{2}{\omega }_{0}t \right){Z}_{x}^{0} {Z}_{y}^{0}-\mathrm{ cos}{\omega }_{0}t \mathrm{sin}{\omega }_{0}t \left({{Z}_{y}^{0}}^{2}\right) \end{array}\right)}_{{\mathcal{R}}_{orb}}dt$$

(161)

Some intermediate calculations, transforming the components in the inertial frame, and assuming $\mathrm{sin}\dot{\alpha }t$ and $\mathrm{cos}\dot{\alpha }t$ constant over one orbit, give:

$${\langle {T}_{GG}\rangle }_{orbit}= \frac{3{\omega }_{0}^{2}}{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]{\left( \begin{array}{c}{\mathrm{cos}\dot{\alpha }t Z}_{y}^{orb}{Z}_{z}^{orb}\\ -{Z}_{x}^{0}{Z}_{z}^{orb}\\ -\mathrm{sin}\dot{\alpha }t{ Z}_{x}^{orb}{Z}_{z}^{orb}\end{array}\right)}_{{\mathcal{R}}_{0}}$$

(162)

This expression can be identified with the following expression:

$$\left(\overrightarrow{{Z}_{orb}}.\overrightarrow{{Z}_{p}}\right){\left(\overrightarrow{{Z}_{p}}\times \overrightarrow{{Z}_{orb}}\right)}_{{\mathcal{R}}_{0}}={Z}_{z}^{orb}{\left(\begin{array}{c}{\mathrm{cos}\dot{\alpha }t Z}_{x}^{orb}+\mathrm{sin}\dot{\alpha }t {Z}_{z}^{orb}\\ {Z}_{y}^{orb}\\ -\mathrm{sin}\dot{\alpha }t {Z}_{x}^{orb}+\mathrm{cos}\dot{\alpha }t {Z}_{z}^{orb}\end{array}\right)}_{\mathcal{R}}\times {\left(\begin{array}{c}\mathrm{sin}\dot{\alpha }t \\ 0\\ \mathrm{cos}\dot{\alpha }t \end{array}\right)}_{\mathcal{R}}$$

(163)

In vectorial notation, this average torque can simply be written:

$${\langle \overrightarrow{{{\varvec{T}}}_{{\varvec{G}}{\varvec{G}}}}\rangle }_{{\varvec{o}}{\varvec{r}}{\varvec{b}}{\varvec{i}}{\varvec{t}}}=\boldsymbol{ }\frac{3{{\varvec{\omega}}}_{0}^{2}}{2}\boldsymbol{ }\left[{{\varvec{I}}}_{{\varvec{z}}}-\left(\frac{{{\varvec{I}}}_{{\varvec{x}}}+{{\varvec{I}}}_{{\varvec{y}}}}{2}\right)\right]\left(\overrightarrow{{{\varvec{Z}}}_{{\varvec{o}}{\varvec{r}}{\varvec{b}}}}.\overrightarrow{{{\varvec{Z}}}_{{\varvec{p}}}}\right)\left(\overrightarrow{{{\varvec{Z}}}_{{\varvec{p}}}}\times \overrightarrow{{{\varvec{Z}}}_{{\varvec{o}}{\varvec{r}}{\varvec{b}}}}\right)$$

(164)

Complementary dynamic analyses are useful to better characterise the evolution of the angular momentum, and in particular its angle ϕ with respect to the orbit normal. Let’s express the angular momentum H in spherical coordinates in the orbital frame (X_orb, Y_orb, Z_orb).

$${\vec{H}}=H{\vec{Z}}_{P}=H{\left(\begin{array}{c}{\text{sin}}\varphi {\text{cos}}\psi \\ {\text{sin}}\varphi {\text{sin}}\psi \\ {\text{cos}}\varphi \end{array}\right)}_{{\mathcal{R}}_{orb}}$$

(165)

We will use the law of derivation in successive frames, ${\mathcal{R}}_{0}$ (X₀, Y₀, Z₀) and ${\mathcal{R}}_{orb}$ (X_orb, Y_orb, Z_orb) to determine the evolution of the angular momentum, assuming that its module remains constant.

$${\left(d\overrightarrow{H}/dt\right)}_{{\mathcal{R}}_{0}}={\left(d\overrightarrow{H}/dt\right)}_{{\mathcal{R}}_{orb}}+ \dot{\alpha } \overrightarrow{{Y}_{0}}\times \overrightarrow{H}={\langle \overrightarrow{{T}_{GG}}\rangle }_{orbit}$$

(166)

$${\left(d\overrightarrow{H}/dt\right)}_{{\mathcal{R}}_{orb}}=H{\left(\begin{array}{c}\dot{\varphi }{\mathrm{cos}}\varphi {\mathrm{cos}}\psi -\dot{\uppsi }{\mathrm{sin}}\varphi {\text{sin}}\psi \\ \dot{\varphi }{\mathrm{cos}}\varphi {\mathrm{sin}}\psi +\dot{\uppsi }{\mathrm{sin}}\varphi {\mathrm{cos}}\psi \\ -\dot{\varphi }{\mathrm{sin}}\varphi \end{array}\right)}_{{\mathcal{R}}_{orb}}+\dot{\alpha }H{\left(\begin{array}{c}{\mathrm{cos}}\varphi \\ 0\\ -{\mathrm{sin}}\varphi {\mathrm{cos}}\psi \end{array}\right)}_{{\mathcal{R}}_{orb}}$$

(167)

$${\langle \overrightarrow{{T}_{GG}}\rangle }_{orbit}=\frac{3{\omega }_{0}^{2}}{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]{\mathrm{cos}}\varphi {\left(\begin{array}{c}{\mathrm{sin}}\varphi {\mathrm{sin}}\psi \\ -{\mathrm{sin}}\varphi {\mathrm{cos}}\psi \\ 0\end{array}\right)}_{{\mathcal{R}}_{orb}}$$

(168)

$$H\left(\dot{\varphi }{\mathrm{cos}}\varphi {\mathrm{cos}}\psi -\dot{\uppsi }{\mathrm{sin}}\varphi {\mathrm{sin}}\psi \right)+\dot{\alpha }H{\mathrm{cos}}\varphi =\frac{3{\omega }_{0}^{2}}{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]{\mathrm{cos}}\varphi {\mathrm{sin}}\varphi {\mathrm{sin}}\psi $$

$$H\left(\dot{\varphi }{\mathrm{cos}}\varphi {\mathrm{sin}}\psi +\dot{\uppsi }{\mathrm{sin}}\varphi {\mathrm{cos}}\psi \right)=-\frac{3{\omega }_{0}^{2}}{2} \left[{I}_{z}-\left(\frac{{I}_{x}+{I}_{y}}{2}\right)\right]{\mathrm{cos}}\varphi {\mathrm{sin}}\varphi {\mathrm{cos}}\psi $$

$$-H\dot{\varphi }{\mathrm{sin}}\varphi -\dot{\alpha }H{\mathrm{sin}}\varphi {\mathrm{cos}}\psi =0$$

(169)

From these three equations and assuming that ${\mathrm{sin}}\varphi \ne 0$ and ${\mathrm{cos}}\psi \ne 0$ we derive the differential equations driving the evolution of the angular momentum in spherical coordinates:

$$\dot{\boldsymbol{\varphi }}=-\dot{\boldsymbol{\alpha }}{\mathbf{cos}}{\varvec{\psi}}$$

$$\dot{{\varvec{\psi}}}=\dot{\varvec{\alpha }}{\boldsymbol{cot}}{\varvec{\varphi} }{\boldsymbol{sin}}{\varvec{\psi}}-\frac{3{{\varvec{\omega}}}_{0}^{2}}{2{\varvec{H}}}\left[{{\user2{I}}}_{{\user2{z}}}-\left(\frac{{{\user2{I}}}_{{\user2{x}}}+{{\user2{I}}}_{{\user2{y}}}}{2}\right)\right]{\boldsymbol{cos}}{\varvec{\varphi }}$$

(170)

Numerical integration confirms that the angular momentum, in its precession motion, describes a spiral around the drifting orbit normal at a precession rate ω_p. Under the assumptions of the previous analyses, there is no drift of the angle phi, only an oscillation. The azimuth angle psi shows the precession motion (Fig. 35).

Fig. 35

Angular momentum precession around the drifting orbit normal due to Gravity Gradient torque

On the contrary, if there was no Gravity Gradient, the angular momentum would remain inertial, would not follow the drift of the orbit pole and the angle $\varphi$ would drift (Fig. 36).

Fig. 36

In the absence of Gravity Gradient torque, the orbit normal drifts away

Appendix C

Tuning of the Earth magnetic dipole model with IGRF-13

This Appendix compares the simplified dipole model presented in Sect. 2.2.1 with the IGRF-13 detailed modeling. Through a proper tuning of the Earth magnetic moment, it justifies its utilization for the analyses. The simplified dipole model follows Eq. (29) and magnitudes are visualized on Fig. 37.

$$\overrightarrow{B}={B}_{eq}\left(-2 {\mathrm{sin}}\theta {\overrightarrow{u}}_{r}+ {\mathrm{cos}}\theta {\overrightarrow{u}}_{\theta }\right)$$

(171)

$${B}_{eq}= \frac{{\mu }_{E}}{{r}^{3}}$$

(172)

Fig. 37

Earth magnetic field (r = 7164 km): dipole model (left) and IGRF-13 model (right)

The IGRF-13 model computes the Earth magnetic field for each latitude and longitude applying the latest IGRF coefficients to Eq. (173). The impact of the magnetic declination and the South Atlantic Anomaly at an altitude of 793 km is visualized in Fig. 37.

$$V\left(r,\theta ,\phi ,t\right)=a\sum_{n=1}^{N}\sum_{m=0}^{n}{\left(\frac{a}{r}\right)}^{n+1}\left[{g}_{n}^{m}\left(t\right){\mathrm{cos}}\left(m\phi \right)+{h}_{n}^{m}\left(t\right){\mathrm{sin}}\left(m\phi \right)\right]{P}_{n}^{m}\left({\mathrm{cos}}\theta \right)$$

(173)

The differences between both models are shown in Fig. 38 along 5 polar orbits of semi major axis 7164 km, distributed over 1 day. The Earth diurnal rotation creates a fair compensation of the orbital differences. On the long term, the mean value of the component ${B}_{y}$ is null, as suggested by the dipole model in polar orbit since the magnetic declination is cancelled by the daily rotation of the Magnetic Pole around the North Pole.

Fig. 38

Comparison between the dipole model and IGRF-13 along 5 polar orbits during 1 day

More precisely, the numerical computation of the square module of the Earth magnetic induction over 1 day permits to fine-tune the Earth magnetic moment ${\mu }_{E}$ to match ${\langle {\Vert \overrightarrow{B}\Vert }^{2}\rangle }_{long term}$ and ${\langle {\Vert {\overrightarrow{B}}_{\perp }\Vert }^{2}\rangle }_{long term}$ which are the relevant parameters for our present application of long term passive magnetic damping.

With the dipole model, all orbits have the same mean value, provided by Eq. (32) reminded hereunder:

$${\langle {\Vert \overrightarrow{B}\Vert }^{2}\rangle }_{orbit}= \frac{5}{2}{B}_{eq}^{2}$$

(174)

With the IGRF model, the orbital mean value changes from one orbit to the next one as shown on Table

Table 11 Mean orbital and daily values of Earth magnetic field square modules by IGRF-13 and the dipole model tuned with ${{\varvec{\mu}}}_{{\varvec{E}}}=7.763.{10}^{15}{{\varvec{T}}.{\varvec{m}}}^{3}$

11 under the column “IGRF-13 B**2” and the relevant data becomes the mean value over 1 day.

With a semi major axis of 7164 km, we get ${\langle {\Vert \overrightarrow{B}\Vert }^{2}\rangle }_{day}=1.114 {10}^{-9} {T}^{2}$ which can be fed in Eq. (32) to obtain the tuned value ${B}_{eq}= 2.111 {10}^{-5} T$ for the dipole model at this altitude.

For any other altitude, the generic parameter to be used is the Earth magnetic moment, ${{\varvec{\mu}}}_{{\user2{E}}}=7.763.{10}^{15}{{\user2{T}}.{\user2{m}}}^{3}$simply derived from Eq. (172):

$${\mu }_{E}= {2.111 {10}^{-5} . (7164.{10}^{3})}^{3}$$

(175)

This is slightly different from the values proposed by reference papers which were targeting different functions, [13] (${\mu }_{E}=7.950.{10}^{15}{ T. m}^{3}$) and [14] (${\mu }_{E}=7.812.{10}^{15}{ T. m}^{3}$).

For our application, the driving parameter for the rate damping is the square module of the component orthogonal to the spin axis since the damping torque according to Eq. (64) is:

$$\overrightarrow{{T}_{SC1D}}=- M {\Vert {\overrightarrow{B}}_{\perp }\Vert }^{2} \overrightarrow{\omega }$$

(176)

The following table shows under the columns “IGRF-13 B_orth**2” the results of the numerical integration over 14 successive orbits of $\frac{1}{{T}_{orbit}} {\int }_{0}^{{T}_{orbit}} {\Vert {\overrightarrow{B}}_{\perp }\Vert }^{2}dt$ for 6 different obliquities $\varphi$ of the spin axis.

The last rows show the corresponding values provided by Eq. (177) for the fine-tuned dipole model:

$$\frac{1}{{T}_{orbit}} {\int }_{0}^{{T}_{orbit}} {\Vert {\overrightarrow{B}}_{\perp }\Vert }^{2}dt\cong -\frac{5}{2}{B}_{eq}^{2}\left(1-0.5 {\mathrm{sin}}^{2}\varphi \right)$$

(177)

IGRF-13 data show a sensible variation from one orbit to the next one but, on the long term, the daily mean error remains below 2%. This validates the utilization of the dipole model with its fine tuned parameter ${\mu }_{E}= 7.763.{10}^{15}{ T. m}^{3}$ for our long term application.