On the Dynamics of Space Debris: 1:1 and 2:1 Resonances

Abstract

We study the dynamics of the space debris in the 1:1 and 2:1 resonances, where geosynchronous and GPS satellites are located. By using Hamiltonian formalism, we consider a model including the geopotential contribution for which we compute the secular and resonant expansions of the Hamiltonian. Within such model we are able to detect the equilibria and to study the main features of the resonances in a very effective way. In particular, we analyze the regular and chaotic behavior of the 1:1 and 2:1 resonant regions by analytical methods and by computing the Fast Lyapunov Indicators, which provide a cartography of the resonances. This approach allows us to detect easily the location of the equilibria, the amplitudes of the libration islands and the main dynamical stability features of the resonances, thus providing an overview of the 1:1 and 2:1 resonant domains under the effect of Earth’s oblateness. The results are validated by a comparison with a model developed in Cartesian coordinates, including the geopotential, the gravitational attraction of Sun and Moon and the solar radiation pressure.

Appendix: On the Derivation of the Cartesian Equations of Motion

We denote by $\theta$ the sidereal time and let $\mathbf {r}$ be the radius vector of the debris with coordinates $(x,y,z)$ and $(X,Y,Z)$ in the quasi-inertial and in the synodic frames introduced in Sect. 2: $\mathbf {r}=x \mathbf {e}_1+y\mathbf {e}_2+z\mathbf {e}_3 = X\mathbf {f}_1+Y\mathbf {f}_2+Z\mathbf {f}_3$. Denoting by $R_3(\theta )$ the rotation matrix of angle $\theta$ around the third axis, the relation between the coordinates is

$$\begin{aligned} \left( \begin{array}{c} x \\ y \\ z \\ \end{array}\right) = R_3(-\theta )\left( \begin{array}{c} X \\ Y \\ Z \\ \end{array}\right) \!. \end{aligned}$$

(6.1)

The equations of motion (2.1) are provided by the sum of the contributions of the Earth’s gravitational influence, including the oblateness effect, the solar attraction, the lunar attraction and the solar radiation pressure. Let us denote by $\nabla _F$ and $\nabla _I$ the gradients in the synodic and quasi-inertial frames:

$$\begin{aligned} \nabla _F\equiv {{\partial }\over {\partial X}}\mathbf {f}_1+{{\partial }\over {\partial Y}}\mathbf {f}_2+{{\partial }\over {\partial Z}}\mathbf {f}_3,\qquad \nabla _I\equiv {{\partial }\over {\partial x}}\mathbf {e}_1+{{\partial }\over {\partial y}}\mathbf {e}_2+{{\partial }\over {\partial z}}\mathbf {e}_3. \end{aligned}$$

The equations (2.1) can be written in the form

$$\begin{aligned} \ddot{\mathbf {r}}&= \ {\mathcal {G}}\ R_3(-\theta )\ \nabla _F \int _{V_\mathrm{E}}{{\rho (\mathbf {r}_\mathrm{p})}\over {|\mathbf {r}-\mathbf {r}_\mathrm{p}|}}\ \mathrm{d}V_\mathrm{E}+ {\mathcal {G}}m_\mathrm{S}\ \nabla _I\Bigl ({1\over {|\mathbf {r}-\mathbf {r}_\mathrm{S}|}}+{{\mathbf {r}\cdot \mathbf {r}_\mathrm{S}}\over {|\mathbf {r}_\mathrm{S}|^3}}\Bigr )\nonumber \\&+\,{\mathcal {G}}m_\mathrm{M}\ \nabla _I\Bigl ({1\over {|\mathbf {r}-\mathbf {r}_M|}}+{{\mathbf {r}\cdot \mathbf {r}_M}\over {|\mathbf {r}_\mathrm{M}|^3}}\Bigr )-C_\mathrm{r}P_\mathrm{r}a_\mathrm{S}^2\ {A\over m}\ \nabla _I\Bigl ({1\over {|\mathbf {r}-\mathbf {r}_\mathrm{S}|}}\Bigr ). \end{aligned}$$

(6.2)

In the synodic frame we can write $(X,Y,Z)=(r\cos \phi \cos \lambda ,r\cos \phi \sin \lambda ,r\sin \phi )$, where $(r,\lambda ,\phi )$ are spherical coordinates with the longitude $0\le \lambda \le 2\pi$ and the latitude $-{\pi \over 2}\le \phi \le {\pi \over 2}$. Following Beutler (2005) and Earth Gravitational Model (2008), $C_{10}=C_{11}=S_{11}=0$ and the values of $C_{21}$ are $S_{21}$ are very small (see Table 1), so that in the Cartesian equations we neglect the contribution of these harmonics. With these remarks we find the following explicit expansion of the Earth’s gravity potential up $n=m=3$:

$$\begin{aligned} V(r,\phi ,\lambda )&\simeq {\mathcal {G}}{{M_\mathrm{E}}\over r}\ \Big [1+\Big ({R_\mathrm{E}\over r}\Big )^2\Big [ {1\over 2}(3\sin ^2\phi -1)C_{20}\\&\qquad \qquad +\,3 \cos ^2\phi \ \Big (C_{22} \cos (2\lambda ) +S_{22} \sin (2\lambda )\Big )\Big ]\\&\qquad \qquad +\, \Big ({R_E\over r}\Big )^3\ \Big [ {1\over 2}\sin \phi (5\sin ^2\phi -3)C_{30}\\&\qquad \qquad +\,\frac{3}{2} \cos \phi (5 \sin ^2 \phi -1)(C_{31} \cos \lambda +S_{31} \sin \lambda )\\&\qquad \qquad +\, 15 \sin \phi (1-\sin ^2 \phi ) \Big (C_{32} \cos (2 \lambda ) +S_{32} \sin (2 \lambda ) \Big )\\&\qquad \qquad +\,15 \cos ^3 \phi \Big (C_{33} \cos (3 \lambda ) +S_{33} \sin (3 \lambda ) \Big )\Big ]\Big ]. \end{aligned}$$

If $A<B<C$ denote the Earth’s principal moments of inertia, we can write $C_{20}=(A+B-2C)/(2M_\mathrm{E} R_\mathrm{E}^2)$ and $C_{22}=(B-A)/(4M_\mathrm{E} R_\mathrm{E}^2)$. The Earth’s gravity potential in the synodic frame becomes

$$\begin{aligned} V(X,Y,Z)&\simeq {\mathcal {G}}{{M_\mathrm{E}}\over r}+{\mathcal {G}}{{M_\mathrm{E}}\over r}\ \Big ({R_\mathrm{E}\over r} \Big )^2\ \Big [C_{20}\Big ({{3Z^2}\over {2r^2}}-{1\over 2} \Big )\\&+\,3C_{22} {{X^2-Y^2}\over r^2}+6S_{22}{{XY}\over r^2}\Big ] +{\mathcal {G}}{{M_\mathrm{E}}\over r}\ \Big ({R_\mathrm{E}\over r}\Big )^3\ \Big [C_{30} \frac{Z}{2r} \Big ({{5Z^2}\over {r^2}}-3\Big )\\&+\, \frac{3}{2} \Big ({{5Z^2}\over {r^2}}-1\Big ) \Big (C_{31} \frac{X}{r} +S_{31} \frac{Y}{r}\Big )\\&\qquad +\, 15 \frac{Z}{r} \Big (C_{32} \frac{X^2-Y^2}{r^2} +S_{32} \frac{2XY}{r^2}\Big )\\&+\,15 \Big (C_{33} \frac{X(X^2-3Y^2)}{r^3} +S_{33} \frac{Y(3X^2-Y^2)}{r^3}\Big )\Big ]. \end{aligned}$$

From this expression and (6.1), we compute the first term of the right hand side of (6.2). This easily leads to the Cartesian equations of motion in the quasi-inertial frame. In (2.2) we give the equations with harmonics up to degree and order two.