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Invariant properties of families of moon-to-earth trajectories

Authors:	J E Lancaster

Institution:	(1) Guidance and Advance Flight Mechanics Dept., McDonnell Douglas Astronautics Company-Western Division, Huntington Beach, Calif., USA

Abstract:	Analytical techniques are employed to demonstrate certain invariant properties of families of moon-to-earth trajectories. The analytical expressions which demonstrate these properties have been derived from an earlier analytical solution of the restricted three-body problem which was developed by the method of matched asymptotic expansions. These expressions are given explicitly to orderµ ^1/2 where is the dimensionless mass of the moon. It is also shown that the inclusion of higher order corrections does not affect the nature of the invariant properties but only increases the accuracy of the analytic expressions.The results are compared with the work of Hoelker, Braud, and Herring who first discovered invariant properties of earth-to-moon trajectories by exact numerical integration of the equations of motion. (Similar properties for moon-to-earth trajectories follow from the principle of reflection). In each instance the analytical expressions result in properties which are equivalent, to orderµ ^1/2, with those found by numerical integration. Some quantitative comparisons are presented which show the analytical expressions to be quite accurate for calculating particular geometrical characteristics. Nomenclature Orbital Elements near the Moon $\bar h$ energy - $\bar l$ angular momentum - semi-major axis - $\bar \varepsilon$ eccentricity - inclination - $\bar \Omega$ argument of node - $\bar \theta$ argument of pericynthion Orbital Elements near the Earth h _e energy - l _e angular momentum - i inclination - argument of node - argument of perigee - t _f time of flight Other symbols $\bar k,\bar m,\bar n,\bar b,\bar c,\bar d$ parameters used in matehing - U a function of the energy near the earth - a function of the angular momentum near the earth - r _p perigee radius - $\bar r_p$ perincynthion radius - $\bar r_n$ radius at node near moon - $\bar v_n$ true anomaly of node near moon - $\bar \sigma ^$ initial angle between node near moon and earth-moon line - $\bar H$ a function ofU, , $\bar r_p$ andi - earth phase angle - dimensionless mass of the moon - U ₀, U₁ U=U* ₀+U ₁ - i ₀, i_1/2, i₁ i=i ₀+µ ^1/2 i _1/2+µ i ₁ - ₀, _1/2, ₁ = ₀+µ ^1/2 i _1/2+µ i ₁ - _p longitude of vertex line - _n latitude of vertex line - R _o,S _o,N _o functions ofU ₀ and - $\bar R_o$ a function ofU ₀, and $\bar r_p$

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