Effect of Geomagnetic Field on Motion of Passive Conductive Satellites

Abstract

In this paper we analyze theoretically the effect of geomagnetic field on rotation of passive conductive satellites of a spherical shape such as Russian-Australian satellite WESTPAC or Russian satellite Larets. The main cause of this effect is the generation of eddy currents and satellite magnetic moment, which interact with the Earth’s magnetic field. The analytical solutions have been derived for orbital and intrinsic magnetic moments of satellite, which result from a translatory motion of an Earth-orbiting satellite and from a satellite spin around axes passing through its center of mass. An arbitrary relation between skin layer depth and satellite radius is assumed. Both a torque acting on satellite in geomagnetic field and the changes in its angular velocity are derived. The satellite orbiting the Earth in a circular orbit with an arbitrary tilt of the orbital plane is studied having regard to rotation of the vector of geomagnetic dipole moment. A characteristic time it takes for the angular velocity of the satellite to decrease exponentially is estimated. Peculiarities of residual irregular vibrations of the satellite angular velocity are examined.

Appendices

APPENDIX 1

The problem under consideration has axial symmetry due to homogeneity of the vector field ${{\partial }_{t}}{{{\mathbf{B}}}_{e}} = {{\dot {B}}_{{ez}}}{\mathbf{\hat {z}}}{\text{.}}$ Here the dot over the symbol denotes the derivative with respect to time. Whence it follows that all the functions depend on variables $r{\text{,}}$$\theta$ and $t$ but are independent of azimuthal angle. In such a case it is suitable to seek for the solution of equation (2) for geomagnetic field perturbations in the form (r < a):

$${{{\mathbf{b}}}_{2}}\left( {{\mathbf{r}},t} \right) = {\mathbf{\hat {r}}}{{b}_{r}}\left( {r,t} \right)\cos \theta - {\mathbf{\hat {\theta }}}{{b}_{\theta }}\left( {r,t} \right)\sin \theta ,$$

(A1.1)

where ${\mathbf{\hat {r}}}$ and ${\mathbf{\hat {\theta }}}$ are the unit vectors while ${{b}_{r}}\left( {r,t} \right)$ and ${{b}_{\theta }}\left( {r,t} \right)$ are unknown functions. Substituting equation (A1.1) for ${{{\mathbf{b}}}_{2}}$ into vector equation (2), taking into account that ${{\partial }_{t}}{{{\mathbf{B}}}_{e}} = {{\dot {B}}_{{ez}}}\left( {{\mathbf{\hat {r}}}\cos \theta - {\mathbf{\hat {\theta }}}\sin \theta } \right)$ and rearranging, we are thus left with the set:

$${{\partial }_{t}}{{b}_{r}} + {{\dot {B}}_{{ez}}} = D\left\{ {\partial _{{rr}}^{2}{{b}_{r}} + \frac{{2{{\partial }_{r}}{{b}_{r}}}}{r} - \frac{{4\left( {{{b}_{r}} - {{b}_{\theta }}} \right)}}{{{{r}^{2}}}}} \right\},$$

(A1.2)

$${{\partial }_{t}}{{b}_{\theta }} + {{\dot {B}}_{{ez}}} = D\left\{ {\partial _{{rr}}^{2}{{b}_{\theta }} + \frac{{2{{\partial }_{r}}{{b}_{\theta }}}}{r} + \frac{{2\left( {{{b}_{r}} - {{b}_{\theta }}} \right)}}{{{{r}^{2}}}}} \right\}.$$

(A1.3)

Equation $\nabla \cdot {\mathbf{b}} = {\mathbf{0}}$ takes form:

$${{\partial }_{r}}{{b}_{r}} + \frac{{2\left( {{{b}_{r}} - {{b}_{\theta }}} \right)}}{r} = 0.$$

(A1.4)

In order to solve the set of equations (A1.2)–(A1.4) we apply a technique used by Ablyazov et al (1988). To do this let us introduce the new unknown functions: $f = {{b}_{r}} - {{b}_{\theta }}$ and $g = {{b}_{r}} + 2{{b}_{\theta }}{\text{.}}$ Rearranging equations (A1.2)–(A1.4) we arrive at

$${{\partial }_{t}}g = D\left( {\partial _{{rr}}^{2}g + \frac{{2{{\partial }_{r}}g}}{r}} \right) - 3{{\dot {B}}_{{ez}}},$$

(A1.5)

$${{\partial }_{t}}f = D\left( {\partial _{{rr}}^{2}f + \frac{{2{{\partial }_{r}}f}}{r} - \frac{{6f}}{{{{r}^{2}}}}} \right),$$

(A1.6)

$${{\partial }_{r}}g + 2{{\partial }_{r}}f + {{6f} \mathord{\left/ {\vphantom {{6f} r}} \right. \kern-0em} r} = 0.$$

(A1.7)

In order to disregard the ball magnetization in the geomagnetic field, the magnetic permeability of the ball is assumed to be unity. From the requirement of the magnetic field continuity at the ball surface it follows that ${{{\mathbf{b}}}_{1}}\left( {a,t} \right) = {{{\mathbf{b}}}_{2}}\left( {a,t} \right){\text{.}}$ Substituting equations (3) and (A1.1) for ${{{\mathbf{b}}}_{1}}$ and ${{{\mathbf{b}}}_{2}}$ into the above relationship, we come to the following boundary conditions

$$g\left( {a,t} \right) = 0,\,\,\,\,f\left( {a,t} \right) = \frac{{3{{\mu }_{0}}{{M}_{{{\text{orb}}}}}\left( t \right)}}{{4\pi {{a}^{3}}}}.$$

(A1.8)

In the absence of any perturbations at the initial time, the initial conditions of the problem are as follows $g\left( {r,0} \right) = f\left( {r,0} \right) = 0.$ The proof of the fact that the set of equations (A1.5)–(A1.7) under the above boundary and initial conditions has a unique solution can be found in (Ablyazov et al., 1988).

The solution of equation (A1.5), which is bounded at $r = 0$ and subjected to the boundary condition (A1.8), can be expressed in a power series of the functions $\sin \left( {{{\pi nr} \mathord{\left/ {\vphantom {{\pi nr} a}} \right. \kern-0em} a}} \right)$ constituting a full set of the orthonormal basis functions for our boundary-value problem:

$$g\left( {r,t} \right) = \frac{1}{r}\sum\limits_{n = 1}^\infty {{{\gamma }_{n}}\left( t \right)\sin \frac{{\pi nr}}{a}} ,$$

(A1.9)

where ${{\gamma }_{n}}\left( t \right)$ are unknown functions. In a similar fashion we can expend the function ${{\dot {B}}_{{ez}}}$ in a power series of the same basis functions despite ${{\dot {B}}_{{ez}}}$ is independent of $r{\text{.}}$ Substituting this power series and equation (A1.9) into equation (A1.5) yields

$$\frac{{d{{\gamma }_{n}}}}{{dt}} + {{\lambda }_{n}}{{\gamma }_{n}} = \frac{{6a{{{\dot {B}}}_{{ez}}}}}{{\pi n}}{{\left( { - 1} \right)}^{n}},\,\,\,\,{{\lambda }_{n}} = \frac{{{{\pi }^{2}}{{n}^{2}}D}}{{{{a}^{2}}}}.$$

(A1.10)

The solution of equation (A1.10) under zero initial conditions is given by

$${{\gamma }_{n}} = \frac{{6a}}{{\pi n}}{{\left( { - 1} \right)}^{n}}\int\limits_0^t {\exp \left\{ {{{\lambda }_{n}}\left( {t{\kern 1pt} ' - t} \right)} \right\}{{{\dot {B}}}_{{ez}}}\left( {t{\kern 1pt} '} \right)dt{\kern 1pt} '} .$$

(A1.11)

Here we have taken into account that ${{\dot {B}}_{{ez}}}$ is not a constant but, generally speaking, a slowly varying function of time. Substituting equation (A1.11) for ${{\gamma }_{n}}$ into equation (A1.9), we obtain the function $g\left( {r,t} \right){\text{.}}$ In order to find other unknown function $f\left( {r,t} \right)$ we use equation (A1.7), whence it follows that

$$\begin{gathered} f\left( {r,t} \right) = - \frac{{{{a}^{2}}}}{{2{{\pi }^{2}}{{r}^{3}}}}\sum\limits_{n = 1}^\infty {\frac{{{{\gamma }_{n}}\left( t \right)}}{{{{n}^{2}}}}\left\{ {\left( {\frac{{{{\pi }^{2}}{{n}^{2}}{{r}^{2}}}}{{{{a}^{2}}}} - 3} \right)} \right.} \\ \left. {^{{^{{^{{^{{}}}}}}}} \times \sin \frac{{\pi nr}}{a} + \frac{{3\pi nr}}{a}\cos \frac{{\pi nr}}{a}} \right\}. \\ \end{gathered} $$

(A1.12)

Substituting equation (A1.12) for $f\left( {a,t} \right)$ into the boundary conditions (A1.8), we come to equation (4) for the effective magnetic moment of the ball.

APPENDIX 2

For the sake of convenience, the derivative in time will be further denoted by a point. From equations (8) and (9) it then follows that the magnetic field components in the coordinate system fixed to the satellite orbit plane have the form:

$$\begin{gathered} \frac{{{{B}_{{ex}}}}}{{{{B}_{*}}}} = \frac{{3{{x}_{m}}{{z}_{m}}}}{{{{R}^{2}}}}\left[ {\cos \alpha \cos \beta \cos {{\varphi }_{e}} + \sin \alpha \sin \beta } \right] \\ - \,\,\frac{{3{{y}_{m}}{{z}_{m}}}}{{{{R}^{2}}}}\cos \alpha \sin {{\varphi }_{e}} + \left( {\frac{{3z_{m}^{2}}}{{{{R}^{2}}}} - 1} \right) \\ \times \,\,\left[ {\cos \alpha \sin \beta \cos {{\varphi }_{e}} - \sin \alpha \cos \beta } \right], \\ {{B}_{*}} = - \frac{{{{\mu }_{0}}{{M}_{e}}}}{{4\pi {{R}^{3}}}},\,\,\,\,\frac{{{{B}_{{ey}}}}}{{{{B}_{*}}}} = \frac{{3{{x}_{m}}{{z}_{m}}}}{{{{R}^{2}}}}\cos \beta \sin {{\varphi }_{e}} \\ + \,\,\frac{{3{{y}_{m}}{{z}_{m}}}}{{{{R}^{2}}}}\cos {{\varphi }_{e}} + \left( {\frac{{3z_{m}^{2}}}{{{{R}^{2}}}} - 1} \right)\sin \beta \sin {{\varphi }_{e}}, \\ \frac{{{{B}_{{ez}}}}}{{{{B}_{*}}}} = \frac{{3{{x}_{m}}{{z}_{m}}}}{{{{R}^{2}}}}\left[ {\sin \alpha \cos \beta \cos {{\varphi }_{e}} - \cos \alpha \sin \beta } \right] \\ - \,\,\frac{{3{{y}_{m}}{{z}_{m}}}}{{{{R}^{2}}}}\sin \alpha \sin {{\varphi }_{e}} + \left( {\frac{{3z_{m}^{2}}}{{{{R}^{2}}}} - 1} \right) \\ \times \,\,\left[ {\sin \alpha \sin \beta \cos {{\varphi }_{e}} + \cos \alpha \cos \beta } \right], \\ \end{gathered} $$

(A2.1)

where ${{\varphi }_{e}} = {{\varphi }_{e}}\left( t \right)$ and the functions ${{x}_{m}}\left( t \right){\text{,}}$${{y}_{m}}\left( t \right)$ and ${{z}_{m}}\left( t \right)$ are given by equation (11). The time derivatives of equations (A2.1) are given by:

$$\begin{gathered} \frac{{{{{\dot {B}}}_{{ex}}}}}{{{{B}_{*}}}} = \frac{3}{{{{R}^{2}}}}\left\{ {\left( {{{{\dot {x}}}_{m}}{{z}_{m}} + {{x}_{m}}{{{\dot {z}}}_{m}}} \right)_{{_{{}}}}^{{}}} \right. \\ \times \,\,\left[ {\cos \alpha \cos \beta \cos {{\varphi }_{e}} + \sin \alpha \sin \beta } \right] + 2{{z}_{m}}{{{\dot {z}}}_{m}} \\ \times \,\,\left[ {\cos \alpha \sin \beta \cos {{\varphi }_{e}} - \sin \alpha \cos \beta } \right] \\ - \,\,\left( {{{{\dot {y}}}_{m}}{{z}_{m}} + {{y}_{m}}{{{\dot {z}}}_{m}}} \right)\cos \alpha \sin {{\varphi }_{e}} - {{\omega }_{e}}\cos \alpha \\ \times \,\,\left[ {{{x}_{m}}{{z}_{m}}\cos \beta \sin {{\varphi }_{e}} + {{y}_{m}}{{z}_{m}}\cos {{\varphi }_{e}}} \right. \\ \left. {\left. { + \,\,\left( {z_{m}^{2} - {{{{R}^{2}}} \mathord{\left/ {\vphantom {{{{R}^{2}}} 3}} \right. \kern-0em} 3}} \right)\sin \beta \sin {{\varphi }_{e}}} \right]} \right\}, \\ \end{gathered} $$

$$\begin{gathered} \frac{{{{{\dot {B}}}_{{ey}}}}}{{{{B}_{*}}}} = \frac{3}{{{{R}^{2}}}}\left\{ {\left( {{{{\dot {x}}}_{m}}{{z}_{m}} + {{x}_{m}}{{{\dot {z}}}_{m}}} \right)\cos \beta \sin {{\varphi }_{e}}_{{_{{}}}}^{{}}} \right. \\ - \left( {{{{\dot {y}}}_{m}}{{z}_{m}} + {{y}_{m}}{{{\dot {z}}}_{m}}} \right)\cos {{\varphi }_{e}} + 2{{z}_{m}}{{{\dot {z}}}_{m}}\sin \beta \sin {{\varphi }_{e}} \\ + \,\,{{\omega }_{e}}\left[ {{{x}_{m}}{{z}_{m}}\cos \beta \cos {{\varphi }_{e}} + {{y}_{m}}{{z}_{m}}\sin {{\varphi }_{e}}} \right. \\ + \,\,\left. {\left. {\left( {z_{m}^{2} - {{{{R}^{2}}} \mathord{\left/ {\vphantom {{{{R}^{2}}} 3}} \right. \kern-0em} 3}} \right)\sin \beta \cos {{\varphi }_{e}}} \right]} \right\}, \\ \end{gathered} $$

(A2.2)

$$\begin{gathered} \frac{{{{{\dot {B}}}_{{ez}}}}}{{{{B}_{*}}}} = \frac{3}{{{{R}^{2}}}}\left\{ {\left( {{{{\dot {x}}}_{m}}{{z}_{m}} + {{x}_{m}}{{{\dot {z}}}_{m}}} \right) \times _{{}}^{{}}} \right. \\ \times \,\,\left[ {\sin \alpha \cos \beta \cos {{\varphi }_{e}} - \cos \alpha \sin \beta } \right] \\ - \,\,\left( {{{{\dot {y}}}_{m}}{{z}_{m}} + {{y}_{m}}{{{\dot {z}}}_{m}}} \right)\sin \alpha \sin {{\varphi }_{e}} + 2{{z}_{m}}{{{\dot {z}}}_{m}} \\ \times \,\,\left[ {\sin \alpha \sin \beta \cos {{\varphi }_{e}} + \cos \alpha \cos \beta } \right] - {{\omega }_{e}}\sin \alpha \\ \times \,\,\left[ {{{y}_{m}}{{z}_{m}}\cos {{\varphi }_{e}} + {{x}_{m}}{{z}_{m}}\cos \beta \sin {{\varphi }_{e}}} \right. \\ \left. { + \,\,\left( {z_{m}^{2} - {{{{R}^{2}}} \mathord{\left/ {\vphantom {{{{R}^{2}}} 3}} \right. \kern-0em} 3}} \right)\left. {\sin \beta \sin {{\varphi }_{e}}} \right]} \right\}, \\ \end{gathered} $$

where

$$\begin{gathered} {{{\dot {x}}}_{m}} = R\left[ {\dot {f}\cos \beta - {{\omega }_{{{\text{orb}}}}}\sin \alpha \sin \beta \sin {{\psi }_{{{\text{orb}}}}}} \right], \\ {{{\dot {y}}}_{m}} = \frac{R}{2}\left[ {\left( {{{\omega }_{{{\text{orb}}}}} - {{\omega }_{e}}} \right)\left( {1 + \cos \alpha } \right)\cos \left( {{{\psi }_{{{\text{orb}}}}} - {{\varphi }_{e}}} \right) + } \right. \\ \left. { + \,\,\left( {{{\omega }_{{{\text{orb}}}}} + {{\omega }_{e}}} \right)\left( {1 - \cos \alpha } \right)\cos \left( {{{\psi }_{{{\text{orb}}}}} + {{\varphi }_{e}}} \right)} \right], \\ {{{\dot {z}}}_{m}} = R\left[ {\dot {f}\sin \beta + {{\omega }_{{{\text{orb}}}}}\sin \alpha \cos \beta \sin {{\psi }_{{{\text{orb}}}}}} \right], \\ \dot {f} = \frac{1}{2}\left[ {\left( {{{\omega }_{{{\text{orb}}}}} + {{\omega }_{e}}} \right)\left( {1 - \cos \alpha } \right)\sin \left( {{{\psi }_{{{\text{orb}}}}} + {{\varphi }_{e}}} \right) - } \right. \\ \left. { - \,\,\left( {{{\omega }_{{{\text{orb}}}}} - {{\omega }_{e}}} \right)\left( {1 + \cos \alpha } \right)\sin \left( {{{\psi }_{{{\text{orb}}}}} - {{\varphi }_{e}}} \right)} \right]. \\ \end{gathered} $$

(A2.3)

Substituting equations (A2.1)–(A2.3) into equation (10), we obtain the torque of Ampere force.