Lesson11:OrbitalTransfersII
10/6/2016
RobinWordsworth
ES160:SpaceScienceandEngineering:
TheoryandApplicaCons
ObjecCves
•  Introduceconceptofsphereofinfluence
•  Studythepatchedconicsapproachto
interplanetarytransfers
•  Understandthephysicalprinciplebehind
flyby/gravityassisttrajectories
SpheresofInfluence
hIp://watchingamerica.com/nrchandelsblad000078.shtml
SpheresofInfluence:Astronomical
hIp://ccar.colorado.edu/asen5050/projects/projects_2012/roberts/roberts_proj.htm
SpheresofInfluence
•  ApproximatemoConofspacecraQis
duetomostinfluen+alcelesCalbody
only
•  DefiniConof‘mostinfluenCal’issubtle
–  Forcebalancedoesnotwork!
–  C.f.MoonaroundEarth
SpheresofInfluence
•  Needtoimagine
spacecraQinorbit
aroundeachbodyin
turn
massofplanet
–  Comparewith
acceleraCondueto
secondbody
•  Earth’sSOIis
–  ~145rE(big)
–  6x10-3AU(small!)
massofSun
Sun-planet
separaCon
SpheresofInfluence
•  Needtoimagine
spacecraQinorbit
aroundeachbodyin
turn
–  Comparewith
acceleraCondueto
secondbody
•  Earth’sSOIis
–  ~145rE(big)
–  6x10-3AU(small!)
Prussing&Conway,OrbitalMechanics
PatchedConics
•  Anengineer’ssoluContoouroldnemesis:the
intractablen-bodyproblemofNewtonian
mechanics
•  PretendaspacecraQ’sorbitisalwaysa
soluContothe2-bodyequaCon(andhencea
conicsecCon)
•  WhatthesoluConisdependsonlyonwhich
sphereofinfluenceyouarein!
PatchedConics
circularorbit
PatchedConics
SUNSOI
circularorbit
EARTHSOI
PatchedConics
SUNSOI
ΔvA
circularorbit
EARTHSOI
PatchedConics
SUNSOI
ΔvA
hyperbolic
escapetrajectory
circularorbit
EARTHSOI
PatchedConics
SUNSOI
ΔvA
???
hyperbolic
escapetrajectory
circularorbit
EARTHSOI
PatchedConics
SUNSOI(~everywhere)
EARTH
PatchedConics
SUNSOI(~everywhere)
EARTH
PatchedConics
SUNSOI(~everywhere)
EARTH
AnExample:Earth-Venustransfer
•  Weareina200km
LEOandwanttopass
500kmoverthe
surfaceofVenus
Steps:
•  Hyperbolicescape
trajectoryfromEarth
•  Hohmanntransferto
Venus
•  Hyperbolicflyby
trajectorypastVenus
with500kmperiapse
VENUS
EARTH
AnExample:Earth-Venustransfer
vA
=1
v2
r
2
1+R
R ⌘ r2 /r1
VENUS
r2
r1
•  Calculateoutbound
Hohmanntransferasbefore
•  R=1AU/0.723AU=1.383
•  v∞,E=2.49km/s
EARTH
v∞,E
AnExample:Earth-Venustransfer
SUNSOI
v∞,E
Nowuseenergy
conservaContoget
perigeespeedof
hyperbolictrajectory
•  vp=11.29km/s
ΔvAisnowassessed
atperigee(itisthe
speedincrease
requiredtoachieve
hyperbolictrajectory
fromcircularorbit)
•  ΔvA=3.5km/s
hyperbolic
escapetrajectory
vp
perigee
rp
circularorbit
EARTHSOI
1 2
v
2 p
µE
1 2
= v1
rp
2
vA = vp
vc
AnExample:Earth-Venustransfer
SUNSOI
VENUSSOI
vp
Exercise:given
rp=500km,
calculateΔvB
requiredfor
orbitalinjecCon
rp
periapsis
(pericytherion)
v∞,V
GravityAssistTrajectories
(PlanetaryFlybys)
•  Avitalpartofmany
interplanetary
missions
•  Firstsuccessful
planetaryflyby:
Mariner2toVenus
(NASA,1962)
•  Firstsuccessfulgravity
assist:Mariner10to
Mercury(NASA,1974)
hIp://nssdc.gsfc.nasa.gov/image/spacecraQ/mariner02.gif
hIp://nssdc.gsfc.nasa.gov/planetary/image/mariner10_labelled.jpg
TheVoyagerFlybys
hEps://vimeo.com/69465942
hIp://solarsystem.nasa.gov/basics/bsf4-1.php
hIp://voyager.jpl.nasa.gov/spacecraQ/goldenrec.html
GravityAssistTrajectories
(PlanetaryFlybys)
•  Firstconsidera
staConaryplanet
•  DirecConof
velocityat
infinitychanges,
magnitudedoes
not
•  SpacecraQKEis
unaltered
v∞,B
v∞,A
GravityAssistTrajectories
(PlanetaryFlybys)
•  Restframe
•  What
happensto
spacecraQ
speedaQer
theflyby?
•  IsenergysCll
conserved?
v∞,B
v∞,B
vE
v2
vE
vE
v∞,A
v∞,A
v1
GravityAssistTrajectories
(PlanetaryFlybys)
v∞,B
Δv
v∞,A
TURNANGLE
vE
v∞,A
v1
v∞,B
Δv
v2
v = 2v1 sin[ /2]
GravityAssistTrajectories
(PlanetaryFlybys)
hIp://solarsystem.nasa.gov/basics/grav/primer.php
GroupDiscussion
Howwouldwecalculatethedelta-vbudgetand
launchsystemrequirementstotransporta2kg
brickfromEarthtothesurfaceofMars?