Perturbations of the First Order in the Motion of Artificial Satellites, Caused by the Flattening of the Earth

The Influence of the Earth’s Oblateness on the Energy Integral and Some Characteristics of a Spacecraft’s Orbit

“…TO THE MOON AND PLANETS Having written out integral (9) for initial point of the trajectory x 0 (r 0 , V 0 , t 0 ) and for some other point x f (r f , V f , t f ), we obtain the ratio (10) Let us apply it to the analysis of the spacecraft departure trajectories from the Earth to the Moon and planets.…”

“…It follows from (10) that the change in the Keplerian energy constants in accurate analysis satisfies the relationship (11) (11a) where i and ω are the inclination and argument of the orbit perigee and u and θ are the latitude argument and true point anomaly.…”

“…Taking into account the oblateness of the Earth, in model ( 4), (5), under the condition h Kf = h Kg , by virtue of integral (10), the initial velocity is determined from the relation ( 16) where Let us provide numerical estimates for the case z 0 = 0. When flying from the Earth to a planet in a geocentric orbit with V ∞ = 3-4 km/s, taking into account the Earth's oblateness in (17), increases initial velocity V 0 in comparison with Keplerian case (16) from ~11.410-11.713 to ~11.413-11.716 km/s, i.e., by ~3 m/s.…”

“…IN THE FIRST APPROXIMATION When analyzing the spacecraft's motion in a hyperbolic and highly elongated elliptical orbit, we used the distance to the center of the Earth as a parameter on the trajectory. In the general case, it is more convenient to use true anomaly θ [9], time t, or average anomaly M [10,11]. We transform relations (11), (12) to use angle θ when determining the change in Keplerian constant Δh K (or semiaxis Δa) as a first approximation.…”