A study of the lunisolar secular resonance $2\dot{\omega}+\dot{\Omega}=0$

The dynamics of small bodies around the Earth has gained a renewed interest, since the awareness of the problems that space debris can cause in the nearby future. A relevant role in space debris is played by lunisolar secular resonances, which might contribute to an increase of the orbital elements, typically of the eccentricity. We concentrate our attention on the lunisolar secular resonance described by the relation $2\dot{\omega}+\dot{\Omega}=0$, where $\omega$ and $\Omega$ denote the argument of perigee and the longitude of the ascending node of the space debris. We introduce three different models with increasing complexity. We show that the growth in eccentricity, as observed in space debris located in the MEO region at the inclination about equal to $56^\circ$, can be explained as a natural effect of the secular resonance $2\dot{\omega}+\dot{\Omega}=0$, while the chaotic variations of the orbital parameters are the result of interaction and overlapping of nearby resonances.