Highlights • The effect of the 2:1 mean motion resonance with the Earth rotation on the critical inclination problem is specified. • A first integral is obtained analytically, and its level curves present the phase structure of the long period motion. • Three types of phase structures are observed, and one of which is first presented.

Abstract A double resonance model is applied to study the long term evolution of a Molniya orbit, which is highly elliptical ($e⩾0.7$), critically inclined ($i\approx 63.4°$), and in the state of the 2:1 mean motion resonance with the Earth rotation. The dynamics of a Molniya orbit can be divided into three kinds: short (12h), intermediate (several years) and long (several centuries) period motions, with the latter two studied in this paper. The $J_2$ and $J_{l2}$ ($l=2\text{,}3\text{,}\dots \text{,}8$) harmonics are modelled, based on a careful selection. The analytic solution for the intermediate period motion is obtained, a first integral, ${\overline{I}}_3$, for the long period motion is derived analytically, and the phase structures are obtained by the level curves of ${\overline{I}}_3$. Three types of the phase structures, depending on the equilibria and stabilities, are observed when the Hamiltonian constant varies. Compared with the near circular 12-h satellite orbits and with the critically inclined orbits without mean motion resonance with the Earth rotation, the features of the Molniya orbits are discussed in detail. It is pointed out that (1) unlike the case of near circular orbits, the $J_{32}$ term does not dominate the 2:1 mean motion resonance problem (intermediate period motion), and that (2) instead of the $J_2^2$ terms, the resonant tesseral harmonics dominate the critical inclination problem (long period motion).