Семинар «Научное наследие профессора Г.Н. Дубошина» К 100-летию со дня рождения профессора Московского Университета Г.Н. Дубошина (19

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 1,5·106 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 L2 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.

INTRODUCTION.
TRAJECTORIES OF DIRECT SPACE FLIGHT AND
BI-ELLIPICAL FLIGHT IN THE EARTH-MOON SYSTEM………... 4
2. MOON-EARTH “DETOUR” FLIGHT IN THE EARTH-MOON-SUN-PARTICLE SYSTEM. SOME NUMERICAL RESULTS ………………8
3. THEORETICAL ANALYSIS OF “DETOUR”FLIGHT …………………12
3.1. EARTH’S GRAVITY EFFECT ON PARTICLE’S ESCAPE……….12
3.2. EARTH’S GRAVITY EFFECT ON PARTICLE’S
ACCELERATION TO HYPERBOLIC MOTION…………………...13
3.3. SUN’S EFFECT ON DECREASING PERIGEE DISTANCE………14
4. CONCLUSIONS……………………………………………………………...15
5. 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.].
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INTRODUCTION. Trajectories of direct space flight - a
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.

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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 . 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.
The first scheme leads to the direct lunar flights, the second one produces Bi-Elliptical lunar flights.
If maximum distance rα from the Earth is large enough, this last scheme is better than the direct flight from energy point of view.
But Sun’s perturbations have to be considered here.
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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.
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Figure 5. Hiten flight
Figure 6. Geocentric Earth-to-Moon trajectory and its
passive prolongation (P1: V∞= 0.4 km/s; P2: 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 (11.05.2001), Es – escape (V∞=0), rmax≈1.47·106 km, F-final point (Hπ =50 km, Δt ≈ 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.
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Algorithm of calculations
The trajectories are defined by integration [Stepan’yants 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 с20, 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 Hπ0 = 100 km, initial selenocentric semimajor axis a0 = 38 455 km, and apolune distance rα ~75 ⋅103 km.
Arc D P1 Es gives elliptic motion. At the point P1 in the flight time Δt ≈ 19 days, aS ≈ 79⋅103 km, and distance ρ ≈ 76⋅103 km.
Es is the escape point. Here, in Δt≈20,6 days, there is zero selenocentric energy, ES=0, ρ≈92 ⋅103 km, Е (Es) gives direction to the Earth. So, there is the escape near translunar libration point L2,
Arc Es P2 P3 gives hyperbolic motion. At the point P2 , for Δt ≈ 21.1 days: ρ ≈ 101⋅103 km, V∞ = 0.15 km s-1. At the point P3 , for Δt ≈ 21.9 days: ρ ≈ 120.2⋅103 km, V∞ = 0.25 km s-1. Then, the spacecraft flies away from both the lunar orbit and the Earth.
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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=2ES =V 2 - 2μM /ρ
versus the time for the initial part of the motion. Here V and ρ 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 ΔV0 ≈ 649 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 a0 = 38455 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 a0 is less, the decreasing in energy will be more, as it is shown at Figure 10.
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Figue 9. Selenocentric energy versus the time for initial part of the Moon-to-Earth detour flight

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Lines H0=100 km correspond to the spacecraft start from the satellite orbit perilune with altitude H0=100 km. Lines H0=0 correspond to the spacecraft start from the Moon surface.
Value Vinf 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 “DETOUR”FLIGHT - a
3.1. EARTH GRAVITY EFFECT ON PARTICLE’S ESCAPE
First, we shall evaluate possibility to have energy increasing ΔES= –E0 for the particle selenocentric motion from initial energy E0 <0 to zero using the evolution theory ( M.L. Lidov 1961, 1962). Suppose eccentricity eS is ~ 1, middle energy Es is - ΔEs/2. Then
ΔES ≈ sign β ((15/2) π μE (μM / aM)3 nM |β|)2/9>0. (3.1)
Here nM is angular velocity of the Moon orbital motion, aM is semi-major axis of Moon’s orbit,
β = cos2 γ sin 2α>0, γ, α are angles of the Moon-Earth vector orientation relative to the particle orbit plane, |β| ≤ 1. Let β be 0.5. Then ΔES≈ 0.096 km2/s2, a0 ≈ 25,600 km. This estimates minimal value of semimajor axis a0 for initial elliptic selenocentric orbit in the Moon-to-Earth detour trajectory.
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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 2451898.5, that is 20.12.2000.0).
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 “DETOUR”FLIGHT - b
3.2. EARTH GRAVITY EFFECT ON PARTICLE’S 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 P2 P3. We use here an approximate linear model, see Figure 12.
The Earth perturbation is
a =aP–aM = -(μE / (rM + ρ)2)((rM + ρ) / (rM + ρ)) + (μE / rM2)(rM / rM). (3.2)
It increases the particle selenocentric energy. Let the Earth-Moon distance rM be сonstant. Then the energy ES is defined by the Moon-particle distance ρ and back:
ES(ρ)-E0=(μE/rM2)(ρ- ρ0)+μE/(rM+ρ) - μE/(rM+ρ0), ES (ρ0)=E0; (3.3)
ρ(ES)=B/2+(B2/4+ rMB)1/2, B=(ES -E0)rM2/ μE + ρ02/(rM +ρ0). (3.4)
Example. Let for the trajectory above in the escape point the energy ES be E0=0, distance ρ be ρ0=91850 km. Then the model (3.2-3.4) gives:
ρ= 102.5 ⋅103 km for V∞=0.15 km/s (point P2, with exact numerical distance ρn=101 ⋅103 km);
ρ= 120.4 ⋅103 km for V∞=0. 25 km/s (point P3, with exact numerical distance ρn=120.2 ⋅103 km).
So, near the translunar libration point L2, the particle can be accelerated by Earth’s gravity from parabolic selenocentric orbit in the escape point Es to the hyporbolic one and move from the Earth.
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Figure 12. A model for the particle selenocentric hyperbolic motion from the Moon (arc Es P2 P3)

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Next, we estimate approximately the Sun gravity effect on the variation Δrπ of the particle orbit perigee distance rπ on the final arc P3 F of the space flight as the orbit revolution.
Suppose that eccentricity e ≈ 1, rπf ≈ 0, middle value rπ ≈ - Δrπ / 2. Then, using the evolution theory
[Lidov 1961, 1962] for the Earth-Sun fixed direction, we have:
Δrπ ≈ sign β ((15 / 2) π (μS /μE) β)2 a7 / aE 6<0. (3.5)
Here μE, μS are the Earth and Sun gravitational parameters; aE is a distance to the Sun; a is semi-major axis of the particle orbit; β= cos2 γ sin 2α<0, γ, α are angles of the Earth-Sun vector orientation relative to the particle orbit plane, |β| ≤ 1. Semi-major axis a that leads to the perigee change Δrπ is
a ≈ [⏐Δrπ⏐ aE 6 / ((15/2) π (μS / μE)β)2]1/7. (3.6)
To evaluate necessary value of semi-major axis a, suppose Δrπ = - 500,000 km, β = - 0,5. Then a ≈ 870,000 km, apogee distance rα ≈ 1.5 million km.
Hence, if the Sun orientation is suitable and the apogee distance is large enough, the perigee distance is decreased to ~ zero, that gives possibility to approach passively the Earth. Numerical calculations confirm this result.
3. THEORETICAL ANALYSIS OF “DETOUR”FLIGHT - c
3.3. SUN GRAVITY EFFECT ON DECREASE OF THE PARTICLE ORBIT
PERIGEE DISTANCE

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Numerical and theoretical studies prove existence of “detour” trajectories
for the Earth-to-Moon passive flight to a lunar satellite orbit with spacecraft’s gravitational capture and
for the Moon-to-Earth passive flight from a lunar satellite orbit with spacecraft’s 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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Capture; Journal of Guidance, Control and Dynamics. Vol. 16. № 4. Pp. 770 - 775.
2. Bello Mora M., F. Graziani, P. Tiofilatto, et al. (2000) A Systematic Analysis On Week Stability
Boundary Transfers To The Moon: Presented at the 51st International Astronautical Congress,
Rio de Janeiro, Brazil, October 2000. Paper IAF-00-A.6.03. 12 p.
3. Biesbroek R., Janin G. (2000) Ways to the Moon? ESA Bulletin. Vol. 103. Pp. 92 - 99.
4. Duboshin G.N. (1963) Celestial Mechanics. Main Problems and Methods (Moscow, USSR:
Nauka, GIPh-ML Publisher.
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Physicheskikh nauk (UPhN), Moscow, USSR. Vol. 63. No. 1a. Pp. 73 – 117.
6. Egorov V.А., and Gusev L.I. (1980) Dynamics of space flights between Earth and Moon
(Moscow, USSR: Nauka Publisher).
7. Gatland K. (1982) The Illustrated Encyclopedia of Space Technology (London: Salamander
Book Ltd.).
8. Hiroshi Yamakawa, et al. (1993) On the Earth-Moon Transfer Trajectory with Gravitational
Capture: Presented at AAS/AIAA Astrodynamics Specialist Conference, Victoria, USA.
Paper AAS 93-633, 20 p.
9. Hohmann W.F. (1925) Die Erreichbarkeit der Himmelskörper (Oldenbourg, Munich).
10. Ivashkin V. V. (2002) On Trajectories of the Earth-Moon Flight of a Particle with its
Temporary Capture by the Moon; Doklady Physics, Mechanics, Vol. 47, No. 11, pp. 825-827.
11. Ivashkin V. V. (2003) On the Earth-to-Moon Trajectories with Temporary Capture of a
Particle by the Moon: Presented at the 54th International Astronautical Congress, held in
Bremen, Germany, September 29 – October 3, 2003. Paper IAC-03-A.P.01, 9 p.
5. REFERENCES - a
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5. REFERENCES - b