Семинар «Научное наследие профессора Г.Н. Дубошина» К 100-летию со дня рождения профессора Московского Университета Г.Н. Дубошина ( гг) Москва, ГАИШ МГУ им. М.В. Ломоносова, 7 декабря 2004 г. Гравитационные возмущения и их роль в формировании лунных траекторий нового типа В.В. Ивашкин Институт прикладной математики им. М.В. Келдыша РАН Миусская пл., 4, Москва, , Россия

Гравитационные возмущения и их роль в формировании лунных траекторий нового типа В.В. Ивашкин 2 АННОТАЦИЯ. В рамках задачи четырех тел (Земля-Луна-Солнце-точка) представлены результаты исследования лунных траекторий Земля-Луна и Луна-Земля нового, «обходного» типа. Данные траектории имеют отлет от Земли на большое расстояние (около 1,5·10 6 км), где под влиянием Солнечных гравитационных возмущений пассивно меняется перигейное расстояние траектории точки от малого значения у Земли до ~ радиуса Лунной орбиты. Это позволяет с помощью гравитационных Земных возмущений осуществить в районе залунной точки либрации L 2 пассивное изменение энергии селеноцентрического движения точки от положительной до нулевой, а затем – до отрицательной, что соответствует движению точки у Луны по орбите спутника Луны, т.е. захвату для полета Земля-Луна и освобождению для полета Луна-Земля.

1.INTRODUCTION. TRAJECTORIES OF DIRECT SPACE FLIGHT AND BI-ELLIPICAL FLIGHT IN THE EARTH-MOON SYSTEM……… MOON-EARTH DETOUR FLIGHT IN THE EARTH-MOON-SUN- PARTICLE SYSTEM. SOME NUMERICAL RESULTS ………………8 3. THEORETICAL ANALYSIS OF DETOURFLIGHT ………………… EARTHS GRAVITY EFFECT ON PARTICLES ESCAPE……… EARTHS GRAVITY EFFECT ON PARTICLES ACCELERATION TO HYPERBOLIC MOTION………………… SUNS EFFECT ON DECREASING PERIGEE DISTANCE………14 4. CONCLUSIONS…………………………………………………………… REFERENCES...………………………………………………….………….16 3 CONTENTS

В своем творчестве Г.Н. Дубошин значительное внимание уделял проблеме вычисления и анализа влияния гравитационных возмущений, в частности, для траекторий полета в системе Земля-Луна. В последнее время были открыты новые классы лунных траекторий, в которых такие возмущения играют особенно большую роль. Кратко опишем их в данном докладе. Исследование космических полетов между Землей и Луной имеют большое значение как для Небесной механики, так и для Космонавтики. Для практически всех полетов, начиная с 1959 г., использовались «прямые» траектории [V.A. Egorov, 1957; V.A. Egorov and L.I. Gusev, 1980; etc.]. 4 I.INTRODUCTION. - a I.INTRODUCTION. Trajectories of direct space flight - a Parking orbit Start Trajectory measurements Correction Deceleration S/C orientation on Lunar Vertical Radioaltimete r switching on Ignition of Retro-engine Soft Landing Figure 1. Scheme of the Luna 9 Mission Схема полета КА Луна-9 для первой мягкой посадки на Луну, а также схема полета КА «Аполлон», первой пилотируемой экспедиции на Луну, приведены здесь для примера.

5 I. INTRODUCTION. - b I. INTRODUCTION. Trajectories of direct space flight - b Figure 2. Scheme of the Apollo Mission For direct flights, trajectories have small enough (several days) time of flight, approach to and departure from the Moon are performed on hyperbolic selenocentric orbits (with velocity at infinity V 1 km/s). This results in the large fuel consumption for spacecraft flights under using these trajectories. It is important to search new low energy lunar flights: a) other schemes; b) Earth-Moon flights with passive capture and Moon-Earth flights with passive escape; c) other types of engines.

I. INTRODUCTION I. INTRODUCTION. Bielliptical Flight in the Earth-Moon System-c In a central field, for flight with a high thrust (impulses), there are two main transfers here: two-impulse Hohmann-Tsander Transfer (Figure 3) and Three-Impulse Bi-Elliptical Sternfeld Transfer, Figure 4. V 2 V 1 rfrf r0r0 O TfTf T0T0 T Figure 3. Two-Impulse Hohmann-Tsander Transfer V 2 V 1 r rfrf V 3 TfTf T0T0 T1T1 T2T2 r0r0 O Figure 4. Three-Impulse Bi-Elliptical Sternfeld Transfer The first scheme leads to the direct lunar flights, the second one produces Bi-Elliptical lunar flights. If maximum distance r from the Earth is large enough, this last scheme is better than the direct flight from energy point of view. But Suns perturbations have to be considered here. 6

New indirect detour Earth- to-Moon flights in frame of the Earth-Moon-Sun-particle system are found recently [Belbruno and Miller 1993; Hiroshi Yamakawa et al 1993; Biesbroek R. and Janin G. (2000); Bellό Mora et al 2000; Koon et al 2001; Ivashkin 2002; etc]. They seem to be similar to Bi-Elliptical flights, but from dynamical point of view they differ from the last ones: ascent of perigee is given by the Sun gravity but not by an impulse and approach the Moon is along the elliptical orbit (with capture) due to the Earth gravity effect. This scheme may be also used for the Moon-to-Earth flight to have a gravitational escape from the Moon attraction [Hiroshi Yamakawa, et al.; V.V. Ivashkin]. Numerical and theoretical analysis has proved existence of these Moon-Earth detour trajectories. 7 Figure 5. Hiten flight Figure 6. Geocentric Earth-to-Moon trajectory and its passive prolongation (P 1 : V = 0.4 km/s; P 2 : V =0.2 km/s; C, Es: V = 0, E=0) Detour Earth-to-Moon flights

2. MOON-EARTH DETOUR FLIGHT IN THE EARTH-MOON-SUN SYSTEM-a Figure 7. The XY view of the geocentric trajectory for detour type: D-departure ( ), Es – escape (V =0), r max 1.47·10 6 km, F-final point (H =50 km, t 113 days ), M - Moon, E – Earth Scheme of Detour Moon-Earth flight These Moon-to-Earth flights in frame of the Earth-Moon-Sun-particle system use first flight from to the Moon orbit and Earth behind the Earth gravity influence sphere and then flight to the Earth. We shall call them by detour flights. From dynamical point of view they differ from the Sternfeld bi-elliptical flights: flight from the Moon is performed along an elliptical orbit due to the Earth effect and descending the perigee is performed by the Sun gravity but not by the impulse. 8 Algorithm of calculations The trajectories are defined by integration [Stepanyants et al] of the particle motion equations in Cartesian nonrotating geocentric- equatorial coordinate system OXYZ. There are taken into account the Earth gravity with its main harmonic с 20, the Moon gravity, and the Sun one. Some Numerical Results. A family of detour trajectories for space flight to the Earth from elliptic orbits of the lunar satellite are found. These trajectories correspond to the spacecraft start from both the Moon surface and the low-Moon elliptic orbit for several positions of the Moon on its orbit. Figure 7 gives a typical detour trajectory.

2. MOON-EARTH DETOUR FLIGHT IN THE EARTH-MOON-SUN SYSTEM-b Figure 8 gives the particle selenocentric motion for initial part of the trajectory. At the point D, on May 11, 2001, for the position of the Moon near its orbit apogee, the spacecraft flies away from the perilune of an initial elliptic orbit with the perilune altitude H 0 = 100 km, initial selenocentric semimajor axis a 0 = km, and apolune distance r ~ km. Arc D P 1 Es gives elliptic motion. At the point P 1 in the flight time t 19 days, a S km, and distance km. Es is the escape point. Here, in t 20,6 days, there is zero selenocentric energy, E S =0, km, Е (Es) gives direction to the Earth. So, there is the escape near translunar libration point L 2, Arc Es P 2 P 3 gives hyperbolic motion. At the point P 2, for t 21.1 days: km, V = 0.15 km s -1. At the point P 3, for t 21.9 days: km, V = 0.25 km s -1. Then, the spacecraft flies away from both the lunar orbit and the Earth. 9 Figure 8. The XZ view for the Moon-to-Earth seleno- centric trajectory of detour type at initial part of the flight

2. MOON-EARTH DETOUR FLIGHT IN THE EARTH-MOON-SUN SYSTEM-c Figure 9 gives the selenocentric energy constant h=2E S =V M / versus the time for the initial part of the motion. Here V and are the selenocentric velocity of the particle and its distance from the Moon. For leaving a 100 km-circular lunar-satellite orbit with a high thrust, the velocity increment is V m/s, that is at about 161 m/s less than for the optimal case of usual direct flight. For a case when spacecraft leaves Moon's surface, the detour trajectory (with a 0 = km again) has approximately the same characteristics as for the indicated case of the start from the lunar satellite orbit. The decrease in the velocity increment is equal to about 156 m/s in this case. If initial semimajor axis a 0 is less, the decreasing in energy will be more, as it is shown at Figure Figue 9. Selenocentric energy versus the time for initial part of the Moon-to-Earth detour flight

11 Lines H 0 =100 km correspond to the spacecraft start from the satellite orbit perilune with altitude H 0 =100 km. Lines H 0 =0 correspond to the spacecraft start from the Moon surface. Value V inf is velocity at infinity V for direct flight: approximately, V =0.8 km/s corresponds to optimal direct flight from the Moon apogee and V =0.9 - to optimal direct flight from the Moon perigee. Figure 10. Decreasing of the velocity impulse for the Moon-Earth detour flight relative to the direct flight depending on the initial semimajor axis Decreasing of the velocity impulse

3. THEORETICAL ANALYSIS OF DETOURFLIGHT - a 3.1. EARTH GRAVITY EFFECT ON PARTICLES ESCAPE First, we shall evaluate possibility to have energy increasing E S = –E 0 for the particle selenocentric motion from initial energy E 0 0. (3.1) Here n M is angular velocity of the Moon orbital motion, a M is semi-major axis of Moons orbit, = cos 2 sin 2 >0,, are angles of the Moon-Earth vector orientation relative to the particle orbit plane, | | 1. Let be 0.5. Then E S km 2 /s 2, a 0 25,600 km. This estimates minimal value of semimajor axis a 0 for initial elliptic selenocentric orbit in the Moon-to-Earth detour trajectory. 12 Hence, the Earth gravity allows increasing the particle energy from initial negative value for elliptical orbit to zero and escape from the Moon attraction. This fits numerical data (see Fig.11, where time t is counted off from the Julian date , that is ). Figure 11. Minimal value of initial semimajor axis depending on the time of start from near-Moon elliptic selenocentric orbit for the Moon-to-Earth detour trajectories

3. THEORETICAL ANALYSIS OF DETOURFLIGHT - b 3.2. EARTH GRAVITY EFFECT ON PARTICLES ACCELERATION TO HYPERBOLIC MOTION Now we approximately analyze the acceleration of the particle motion with respect to the Moon from the zero energy to a positive one for a hyperbolic motion with velocity at infinity V 0.15 – 0.25 km/s on the following short arc Es P 2 P 3. We use here an approximate linear model, see Figure 12. The Earth perturbation is a =a P –a M = -( E / (r M + ) 2 )((r M + ) / (r M + )) + ( E / r M 2 )(r M / r M ). (3.2) It increases the particle selenocentric energy. Let the Earth-Moon distance r M be сonstant. Then the energy E S is defined by the Moon-particle distance and back: E S ( )-E 0 =( E /r M 2 )( - 0 )+ E /(r M + ) - E /(r M + 0 ), E S ( 0 )=E 0 ; (3.3) (E S )=B/2+(B 2 /4+ r M B) 1/2, B=(E S -E 0 )r M 2 / E /(r M + 0 ). (3.4) Example. Let for the trajectory above in the escape point the energy E S be E 0 =0, distance be 0 =91850 km. Then the model ( ) gives: = km for V =0.15 km/s (point P 2, with exact numerical distance n = km); = km for V =0. 25 km/s (point P 3, with exact numerical distance n = km). So, near the translunar libration point L 2, the particle can be accelerated by Earths gravity from parabolic selenocentric orbit in the escape point Es to the hyporbolic one and move from the Earth. 13 Figure 12. A model for the particle selenocentric hyperbolic motion from the Moon (arc Es P 2 P 3 )

1414 Next, we estimate approximately the Sun gravity effect on the variation r of the particle orbit perigee distance r on the final arc P 3 F of the space flight as the orbit revolution. Suppose that eccentricity e 1, r f 0, middle value r - r / 2. Then, using the evolution theory [Lidov 1961, 1962] for the Earth-Sun fixed direction, we have: r sign ((15 / 2) ( S / E ) ) 2 a 7 / a E 6

1515 Numerical and theoretical studies prove existence of detour trajectories for the Earth-to-Moon passive flight to a lunar satellite orbit with spacecrafts gravitational capture and for the Moon-to-Earth passive flight from a lunar satellite orbit with spacecrafts gravitational escape from lunar attraction. They require less fuel consumption, although have a long enough flight time and need more exact navigation support. 4. CONCLUSIONS

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