lunar transfer
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Seminar 7 !Lunar Missions, Science, & Astronauts!
Lunar Transfer Trajectories!FRS 112, Princeton University!
Robert Stengel"
Copyright 2015 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/FRS.html!
Missions to the Moon!Apollo 8, 9!Apollo 10!Apollo 11!Apollo 12!
Lunar Transfer Trajectories!Understanding Space, Sec 7.2, 7.3!
Modern Spacecraft Dynamics & Control, Sec 3.5!1!
A Man on the Moon!
2!
Apollo 10!May 1969 !
2nd manned flight to the Moon"No landing intended"
Apollo 9!March 1969 !
Earth-orbit test of Lunar Module, rendezvous, and docking"
3!
4!
Apollo 10 Mission"
On the Way to the Moon"
5!
On the Way to the Moon"
6!
Lunar Landing Research/Training Vehicles (LLRV, LLTV)"
7!
https://www.youtube.com/watch?v=mBlNfFcV6ns!
July 16, 1969 ! July 20, 1969 !Apollo 11 !
8!
Apollo 11"!" Powered descent, LM flip at 46 Kft"!" Maneuvering jets – Reaction Control System (RCS)"!" 1202 Alarm: Rendezvous Radar"!" Switch to manual control"
9!
Cause of 1202 Alarm"
10!
11!
Descent to Surface from Circular Orbit"
12!
Powered Descent to Surface"
13!
Phases of Powered Descent"
14!HBO: “From the Earth to the Moon”, Part 6!
Apollo 12"November 14-24, 1969"
15!
Pete Conrad, Surveyor 3, and LM!
Apollo 13"April 11-17, 1970!
16!
Apollo 13 Trajectory"
17!
Apollo 14!January 1971!
Apollo 15!July 1971!
Apollo 16!April 1972!
18!
Apollo 18-20: Cancelled!
Apollo 17, December 1972!
19!
Lunar Transfer Trajectories!
20!
Hyperbolic Encounter with
a Planet "•" Trajectory is
deflected by target planet s gravitational field"
•" Velocity w.r.t. Sun is increased or decreased"
Kaplan!
! : Miss Distance, km" : Deflection Angle, deg or rad
21!
Effect of Target Planet s Gravity on Probe s Sun-Relative Velocity"
Deflection – Velocity Reduction!
22!
Effect of Target Planet s Gravity on Probe s Sun-Relative Velocity"
Deflection – Velocity Addition!
23!
Earth Escape Trajectory "#v to increase speed to escape velocity"
Velocity required for transfer at sphere of influence"
24!
Planet Capture Trajectory "Hyperbolic approach to planet’s sphere of influence"
#v to decrease speed to circular velocity"
25!
Earth-Moon-Spacecraft Dynamics"
Equations of motion include inverse-square gravitational equations for both Earth and Moon!
26!
Fixed-Earth Co-planar (2-D) Model!
!vx t( ) = !µE xE t( ) rE3 t( )! µM "xM t( ) "rM 3 t( )!vy t( ) = !µE yE t( ) rE3 t( )! µM "yM t( ) "rM 3 t( )
!xE t( ) = vx t( )!yE t( ) = vy t( )
!xM = xE ! xMoon!yM = yE ! yMoon!rM = !xM
2 + !yM2
Earth-Moon Sphere of Influence"
Actual “sphere” of influence is not a sphere (Battin, 1964)!
rSI ! rEarth!MoonmMoon
mEarth
"#$
%&'
2 5
! 66,100 km
... but rSI "14rEarth!Moon
27!
Lunar Trajectory "
Direct transfer orbit may appear elliptical wrt Earth until Moon encounter"
Travel time reduced for parabolic or hyperbolic transfer"
28!
29!
Lunar Impact Trajectory "
30!http://en.wikipedia.org/wiki/Ranger_program!
Lunar Fly-By Trajectory"(from CoPlanarTraj.m) "
31!
Inertial Reference Frame! Reference Frame Rotating with the Moon!
Michielsen Chart for Lunar Encounter "(Kaplan, Modern Spacecraft Dynamics and Control, 1976, Sec 3.5)"
•" Orbital velocity of the moon (wrt earth), v ~ 1 km/s"
•" Probe s approach velocity specified by transfer trajectory"
•" Probe s approach deflection angle specified by moon-relative hyperbolic trajectory"
•" Vector triangles for probe s departure velocity vector"
•" Transfer time on plot"•" Earth intercept zone
connotes return to earth without thrusting maneuver"
•" Earth escape also possible"
vMoon! vdepart!
vapproach!
v"+!
v"–!
#!
32!
Apollo Free-Return Trajectory"Trajectories to lunar orbit or landing typically
pass in front of the moon"Thrusting maneuver on the far side required for
lunar orbit or landing"With proper approach velocity, trajectory is
deflected to “Figure 8” pattern for free return "
Kaplan! 33!
Apollo Free-Return Trajectory"
34!
Restricted 3-Body Problem!
35!
Lagrange (or Libration) Points "
Joseph –Louis Lagrange!(1736-1813)!
“Gravity Wells” of the Rotating Earth-Moon
System!
Moon!Earth!
•" Earth and Moon in pre-defined circular orbit, with mean motion, n = 2"/Period!
•" x-y Coordinates rotate with angular rate, n!
•" Contours of constant energy of spacecraft!
36!
Expression of Energy in a Rotating 2-Body Gravitational Field"
Jacobi Constant!
37!
Lagrange (or Libration) Points "!" Lagrange Points are fixed in rotating coordinate frame"!" Spacecraft can orbit about a Lagrange Point"!" Orbits about L1, L2, and L3 are unstable"!" Orbits about L4 and L5 are stable"
38!
Sun-Earth Lagrange Points "
39!
Earth-Moon Lagrange Points "
40!
!Sun-Earth "Earth-Moon"L1 to m !1,501,557 km !64,499 km!L2 to m !1,491,557 km !58,006 km!L3 to M !149,599,737 km !381,678 km!
Low-Thrust/Energy Transfers!
41!
Unmanned Lunar Flight Revisited"
42!
Fast, high-thrust (~impulsive)
trajectory!Slow, low-thrust
trajectory!
Slow, high-thrust (~impulsive)
trajectory!
Ion/Plasma Thrusters"
43!
Engine" Propellant" Required power"Specific impulse" Thrust"kW" s" mN"
NSTAR" Xenon! 2.3! 3,300 to 1,700! 92 max!NEXT[" Xenon! 6.9! 4,300! 236 max!HiPEP" Xenon! 20–50! 6,000–9,000! 460–670!
Hall effect" Xenon! 25! 3,250! 950!FEEP" Liquid Cesium! 6#10$5–0.06! 6,000–10,000! 0.001–1!
VASIMR" Argon! 200! 3,000–12,000! ~5,000!DS4G" Xenon! 250! 19,300! 2,500 max!
Variable Specific Impulse Magnetoplasma Rocket (VASIMR) "
44!
Propellant"Required
power"Specific impulse" Thrust"
kW" s" mN"Argon! 200! 3,000–12,000! ~5,000!
DAWN Spacecraft"
45!
Engine" Propellant" Required power" Specific impulse" Thrust"kW" s" mN"
NSTAR" Xenon" 2.3" 3,300 to 1,700" 92 max"
Unmanned Lunar Flight Revisited"
“Weak Stability” Regions!
Space probe orbits affected by subtle gravitational effects!Propellant savings (Belbruno et al, 1990)!
Missions salvaged or re-purposed (Hiten, HGS-1, ARTEMIS)!Long-duration maneuvers!
Earth! Moon!
46!
Hiten Trajectory to Lunar Orbit"
47!
Two satellites re-purposed to orbit the Moon!48!
49!
GRAIL Spacecraft Used Sun-Earth L1 Point for Low-Energy Transfer
to Lunar Orbit"
Sun" Earth"
50!
Next Time:!!" Building Spacecraft & Launch Vehicles: "
!" [Chariots for Apollo] Ch 4 to 8!!" [Stages to Saturn], Ch 3 to 8!
!" Space Vehicle Design: %[Understanding Space] Ch 11, Sec 13.3, 13.4!
51!
Supplemental Material!
52!
Hyperbolic Orbits"
r = p1+ ecos!
=a 1" e2( )1+ ecos!
cos! = 1ea 1" e2( )
r"1
#
$%%
&
'((
Asymptotic Value of True Anomaly!
! r"#$ %$$!#
!# = cos&1 & 1e
'()
*+,
Polar Equation for a Conic Section"
53!
Hyperbolic Orbits"
h = Constant = v!"
= µp = µa 1# e2( ) = µ2 e2 #1( )v!
2
e = 1+ 2h2E
µ2 = 1+ v!4"2
µ2
rp = a 1! e( ) = µv"2 e!1( )
e = 1+rpv!
2
µ"
#$
%
&'
Angular Momentum!
Eccentricity!
Eccentricity!
Perigee Radius!
54!
Mercury MESSENGER Fly-By Trajectories"
Mercury MESSENGER Mission!https://www.youtube.com/watch?v=otF2FjpCyZk! 55!
Cassini Fly-by Trajectories "
56!
Early Lunar Spacecraft "•"Mission"
–" Scientific discovery"–" Preparations for human voyages to the moon"
•"Robotic exploration of the moon"–" http://en.wikipedia.org/wiki/Robotic_exploration_of_the_Moon"
Lunakhod!Ranger!(pre-Apollo)!
57!
Lunar Spacecraft "
•"Lunar spacecraft launched by US, ESA, Russia, and Japan"
•"Future launches planned by China, Germany, and India"–" http://en.wikipedia.org/wiki/List_of_future_lunar_missions"
Lunar!Prospector (1998)!
SMART-1!(2003, solar-powered ion engine, Isp = 1,640 s)!
58!
Weak Stability Region Trajectories"•" Use of Earth-Moon-Sun gravitational effects to
produce low-energy maneuvers"•" Very long transfer times"•" Ballistic lunar transfer (BLT)"
Marsden, Ross!
Belbruno!
59!
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