The aim of this work is to analyze the orbital evolution of the mean eccentricity given by the Two-LineElements (TLE) set of the Molniya satellites constellation. The approach is bottom-up, aiming at a synergy between the observed dynamics and the mathematical modeling. Being the focus the long-term evolution of the eccentricity, the dynamical model adopted is a doubly-averaged formulation of the third-body perturbation due to Sun and Moon, coupled with the oblateness effect on the orientation of the satellite. The numerical evolution of the eccentricity, obtained by a two-degree-of-freedom time dependent model assuming different orders in the series expansion of the third-body effect, is compared against the mean evolution given by the TLE. The results show that, despite being highly elliptical orbits, the second order expansion catches extremely well the behavior. Also, the lunisolar effect turns out to be non-negligible for the behavior of the longitude of the ascending node and the argument of pericenter. Finally, a frequency series analysis is proposed to show the main contributions that can be detected from the observational data.

An illustrative example of Molniya orbit ground track. Two 12-hr orbits are displayed to span a full day. The nominal orbital parameters are assumed, for illustrative scope, according to the template of Fortescue (1995). In particular, for our specific choice, we set the relevant geometric orientation parameters (i, ω) = (63.4^o, 270^o), with orbit scale length parameters (in km) (a, h_p, h_a)_km = (26560, 1000, 39360). This implies an eccentricity e = 0.72 and a period P = 720 min. Note the extremely asymmetric location of the ascending (“AN”) and descending (“DN”) nodes, due to the high eccentricity of the orbit, and the perigee (“P”), always placed in the Southern hemisphere. Along the two daily apogees (“A”), the satellite hovers first Russia and then Canada/US. The visibility horizon (aka the “footprint”) attained by the Molniya at its apogee over Russia, is above the displayed yellow line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Semi-major axis (left; km), eccentricity (middle) and pericenter altitude (right; km) mean evolution from the TLE data of Molniya 2-09, 2-10, 2-13, 1-36, 3-13, 1-69 (#1, 2, 4, 9, 15, 25 of Table 1). On the right, the time is displayed in decimal year for the sake of clarity; also, the black horizontal line highlights 250 km of altitude.

Eccentricity evolution obtained by assuming different levels of third-body perturbation of (e, i, Ω, ω), compared against the TLE evolution for some specific examples. More details in the text. Top: orbits #1 and #2: middle: #7 and #13; bottom: #15 and #19. All the other orbits are shown in the “SUPPLEMENTARY MATERIAL 2”. The TLE evolution is displayed in cyan; the evolution obtained by applying model 1 in green; the evolution obtained by applying model 2 in red; the evolution obtained by applying model 3 in black; the evolution obtained by applying model 4 in yellow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Longitude of the ascending node (left) and argument of pericenter evolution (right) evolution obtained by assuming different levels of third-body perturbation of (e, i, Ω, ω), compared against the TLE evolution for some specific examples. More details in the text. Top: #2; middle: #7; bottom: #15. The TLE evolution is displayed in cyan; the evolution obtained by applying model 1 in green; the evolution obtained by applying model 2 in red; the evolution obtained by applying model 3 in black. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

On the left, in cyan the eccentricity evolution of the TLE for Molniya 2-10 (orbit #2) and the ones obtained by numerical propagation choosing two different initial conditions (black Table 1 and magenta Table 2). On the right, a similar experiment performed for Molniya 3-24 (orbit #23). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Propagation of 11 neighboring initial conditions, given the same initial epoch and using model 3. The initial conditions have been taken from the propagation of the initial condition in Table 1, Table 2 for orbit #1 (left) and #2 (right), respectively, by selecting the points x_n = x(t_n), t_n = t_o + nΔt, n ∈ {−K, ..., K}, Δt = 1 day from the nominal trajectory. Such neighboring initial conditions are not discernible the one from the other. The divergence among different orbits can be appreciable, but very slightly, only towards of the end of the interval of propagation of Molniya 2-10 on the right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Example of the results obtained by applying a Lomb–Scargle procedure to the eccentricity series. From left to right: Molniya 2-09, 2-10, 1-29 (orbit #1, 2, 3, respectively). In red, the dominant terms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)