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C.3 Moon-Phase Parameters: v, EMP Angle at SoI Exit, and the B-Plane

C.3 Moon-Phase Parameters: v, EMP Angle at SoI Exit, and the B-Plane

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30

50



40



50



60



70



80



C



D



B



A′



A



60

70

80

90

OUTWARD TIME OF FLIGHT T0 (HOURS)



ANALYTIC PROGRAM

EXACT PROGRAM



50



C



D



B



A′



A



60

70

80

90

OUTWARD TIME OF FLIGHT T0 (HOURS)



CASE II



Fig. C.3 Probe-moon-earth angle versus the outward time of flight for various positions of the moon in its orbit



PME ANGLE (DEGREES)



CASE I



360

Appendix C Additional Penzo Parametric Plots



5000



5000



3000



4000



VELOCITY V∞ (FPS)



60



70

80

90

TIME OF FLIGHT T0 (DEGREES)



I



II



POSITION C



I−A



I−A′



II−A



II−A′



ANALYTIC PROGRAM

EXACT PROGRAM



60



70

80

90

TIME OF FLIGHT T0 (DEGREES)



I



II



POSITION D



I



II



POSITION B



Fig. C.4 Hyperbolic excess velocity versus the outward time of flight for fixed outward-phase inclinations and various positions of the moon in its orbit



3000



4000



VELOCITY V∞ (FPS)



POSITIONS A AND A′



Appendix C Additional Penzo Parametric Plots

361



30

50



40



50



60



70



80



C



B



D



A′



A



60

70

80

RETURN TIME OF FLIGHT Tr (HOURS)



ANALYTIC PROGRAM

EXACT PROGRAM



90



50



70



80

RETURN TIME OF FLIGHT Tr (HOURS)



60



RETURN INCLINATION IMR = 90 DEGREES



90



C



B



D



A′



A



Fig. C.5 Earth-moon-probe angle versus the return time of flight for various positions of the moon in its orbit at the spacecraft’s exit from the SoI



EMP ANGLE (DEGREES)



RETURN INCLINATION IMR = 0 DEGREES



362

Appendix C Additional Penzo Parametric Plots



Appendix C Additional Penzo Parametric Plots



363



RETURN INCLINATION IMR = 180Њ



80



EMP ANGLE (DEGREES)



70



60



A



50



A′

D



40



B

C



30

50



60

70

80

RETURN TIME OF FLIGHT Tr (HOURS)



90



Fig. C.6 Earth-moon-probe angle versus the return time of flight for various positions of the moon

in its orbit at the spacecraft’s exit from the SoI (continued)



return inclinations are considered. Note that the EMP angle increases as the return

inclination rotates from ccw to cw.

Finally, traces of the (impact) B-vector measured in the (impact) B-plane are

presented in Fig. C.7. The B-plane is defined to be perpendicular to the incoming

hyperbolic asymptote. The horizontal or B•T axis lies in the moon’s orbit plane and

is directed positively opposite the moon’s direction of motion. The vertical or B•R

axis is directed positively below the moon’s orbit plane. The origin is fixed to the

center of the moon. For all circumlunar trajectories considered here, the asymptote

is inclined only Ỉ10 to the moon’s orbit plane. The PME angle, which indicates

the position of the asymptote wrt the earth-moon line, also indicates the position of

the impact parameter plane. Since this angle varies greatly with time of flight,

outward inclination, and position of the moon, the impact parameter plane is

oriented differently from the earth-moon line for each point in Fig. C.7.

Both graphs in Fig. C.7 are drawn for the maximum distance of the moon. The

outward flight time and the return inclination wrt the moon’s orbit plane are



364



Appendix C Additional Penzo Parametric Plots

CASE I − c



−0.5

270Њ



75 HOUR

70 HOUR



B·R (107 FEET)



270Њ



65 HOUR

60 HOUR



0







180Њ



90Њ

90Њ



0.5

−2.5



−2.0



−1.5

−1.0

B·T (107 FEET)



0



CASE II − c



−0.5

80 HOUR



B·R (107 FEET)



−0.5



75 HOUR



70 HOUR



270Њ



65 HOUR



60 HOUR







0





180Њ



180Њ

90Њ



0.5

−2.5



−2.0



−1.0

−1.5

B·T (107 FEET)



−0.5



0



Fig. C.7 Components of the impact vector B for constant outward times of flight



indicated. The larger magnitude of the impact parameter for a fixed time of flight

implies a greater pericynthion distance. A greater time of flight produces a greater

impact parameter since the incoming asymptote must be further from the moon for

lower moon-centered energies and the same turning of the spacecraft’s velocity

vector.



Answers to Selected Exercises



Chapter 1

1.3.

1.5.

1.6.



α, β

(2.0164, À1.9377, 18.4981)

(17.9610, 4,7662, 2.1651) (km)



Chapter 2

2.9.



(a)

(b)

2.11. (e)

2.16. (a)



(b)

(c)



(d)



(e)



ellipse

286.83

(π/2 À e)/(2π)

β0 ¼ 45.17

r_ ¼ 1:41045 km=s

r_ θ ¼ 1:40229 km=s

radial component vector ¼ (À1.17271992, À0.72798589, 0.29004357)

transverse component vector ¼ (0.77502268, À1.02438770, 0.56248357)

a ¼ 970375.5 km

e ¼ 0.8495

p ¼ 540145.94 km

rp ¼ 292046.76 km

ra ¼ 3589455.02 km

τ ¼ 31.92 days

θ ¼ 168.57

E ¼ 141.35

M ¼ 110.94

tp ¼ 9.8371 days

Time to next periapsis ¼ 22.0833 days



# Springer International Publishing Switzerland 2015

G.R. Hintz, Orbital Mechanics and Astrodynamics,

DOI 10.1007/978-3-319-09444-1



365



366



Answers to Selected Exercises



Chapter 3

3.1. (c) E ¼ À12:65 km2 =s2

a ¼ 15,759 km

p ¼ 11430.6 km

h ¼ 67500 km2/s

3.3. (a) e ¼ 0.34

θ0 ¼ 110

E ¼ À28:4417km2 =s2

a ¼ 7007.3 km

rp ¼ 4624.8 km

ra ¼ 9389.8 km

E ¼ 90

(b) β ¼ 30

v ¼ 7.91 km/s

3.9. (a) 6.305 km/s

(b) 116.07

3.14. (a) e ¼ 0.1179

θ ¼ 119.48

h ¼ 5.338 Â 109 km2/s

rp ¼ 1.9207 Â 108 km

a ¼ 2.1775 Â 108 km

τ ¼ 641.41 days

(b) e ¼ 0.1179

θ ¼ 240.52

h ¼ 5.338 Â 109 km2/s

rp ¼ 1.9207 Â 108 km

a ¼ 2.1775 Â 108 km

τ ¼ 641.41 days

3.16. ΔvSOI ¼ 0.624 km/s

3.20. (a) Thrust ¼ 96,500 N

(b) Δv ¼ 0.6152 km/s

3.21. Δm ¼ 9924.72 kg

3.22. Δm ¼ 850.9 kg

Chapter 4

i ¼ 26.8 , Ω ¼ 187.2 , ω ¼ 341.9

(a) θ ¼ 207.23

(b) r ¼ (À174663897.0, 74685964.6, 49660154.5) km

v ¼ (À5.44426, À21.66724, À3.38179) km/s

4.12. (a) a ¼ 5.112 AU

e ¼ 0.8045



4.8.

4.9.



Answers to Selected Exercises



4.20. (a) 48,214,824 km

(b) 112,378 km

(c) 43,317 km

Chapter 5

5.3.

5.7.



(a) i ¼ 63.43 or 116.57

(b) 63.43 < i < 116.57

2.0 Â 10À5



Chapter 6

6.1.

6.2.

6.5.



ΔVTOTAL ¼ 3.932 km/s

(b) ϕ ¼ 1.76 rad ¼ 100.9

(c) ϕ ¼ 0.725 rad ¼ 41.5

(b) ΔVTOTAL ¼ 0.2428 m/s



Chapter 7

7.1.



(b)







∂τ 

∂VM θ¼0





∂τ 

∂VM 



¼ 5:133 min=mps



θ¼180



¼ 0:442 min=mps



(a) δ ¼ 80.3

(c) rp ¼ 2,292 km

7.9. (a) rp ¼ 75,121 km

ra ¼ 700,546.2 km

τ ¼ 27.82 days

(b) Δm ¼ 186.64 kg

7.13. (c) hp ¼ 639 km

Tr ¼ 80 h

7.15. (a) Az ¼ 90

(b) ireq ¼ 65

7.7.



367



Acronyms and Abbreviations



AAS

ACM

AD

AGA

AIAA

aka

APS

A Train

AU

ATV

AutoNav

BMW

C3, C3

CA

ccw

cw

cm

CM

CNES

Co.

CONAE

CSM

CW

EDT

DLA

DSM

DOF

DPS

DS1

DSM

DSN



American Astronomical Society

Attitude correction maneuver

Arrival date

Aerogravity assist

American Institute of Aeronautics and Astronautics

Also known as

Ascent Propulsion System

Afternoon constellation of spacecraft

Astronomical unit

Automated Transfer Vehicle

Autonomous (optical) navigation

Text entitled Fundamentals of Astrodynamics by Roger R. Bate,

Donald D. Mueller, and Jerry E. White

Launch energy

Closest approach

Counterclockwise

Clockwise

Center of Mass

Command Module

Centre National d’Etudes Spatiales (French Space Agency)

Company

Comision Nacional de Actividades Espaciales (Space Agency of

Argentina)

Command and Service Module

Clohessy-Wiltshire (equations or reference frame)

Eastern Daylight Time

Declination of launch azimuth

Deep Space Maneuver

Degrees of freedom

Descent Propulsion System

Deep Space 1 (spacecraft or space mission)

Deep Space Maneuver

Deep Space Network



# Springer International Publishing Switzerland 2015

G.R. Hintz, Orbital Mechanics and Astrodynamics,

DOI 10.1007/978-3-319-09444-1



369



370



ECI

EDL

EGA

EI

EMP

EOM

ERT

ESA

ET

FAQ

FF

FOM

FPVS

FT

GEO

Glonass

GNC, GN&C

GPS

GRB

GSFC

GTO

HCA

HEO

IAU

IEEE

IERS

I/P

ISS

J2

JAXA

JD

JPL

KI

KII

KIII

KE

LAT

LD

LEO

LEM

LM

LOI



Acronyms and Abbreviations



Earth-centered inertial

Entry, descent, and landing

Earth gravity assist

Entry interface

Earth-moon-probe (angle at exit of the moon’s sphere of influence)

Equation of motion

Earth-received time

European Space Agency

Ephemeris Time

Frequently asked question

Formation flying

Figure of merit

Flight Plane Velocity Space

Flight time

Geostationary orbit, geosynchronous orbit

Global Navigation Satellite System aka Global Orbiting

Navigation satellite system

Guidance, navigation, and control

Global Positioning System (aka NAVSTAR)

Gamma ray bursts

Goddard Space Flight Center

Geosynchronous transfer orbit

Heliocentric angle

High earth orbit, highly elliptical orbit

International Astronomical Union

Institute of Electrical and Electronics Engineers

International Earth Rotation Service (formerly the Bureau International L’Heure, BIH)

Interplanetary

International Space Station

Oblateness coefficient in spherical harmonic expansion

Japanese Aerospace Exploration Agency (Japanese Space

Agency)

Julian Date

Jet Propulsion Laboratory

Kepler’s first law

Kepler’s second law

Kepler’s third law

Kinetic energy

Latitude

Launch date

Low earth orbit

Lunar Excursion Module

Lunar Module

Lunar Orbit Insertion



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