Orbital'maneuvers'

Orbital'Maneuvers'

Chapter'6' Maneuvers'

• We'assume'our'maneuvers'are'“instantaneous”'' • ΔV'changes'can'be'applied'by'changing'the' magnitude'“pump”'or'by'changing'the'direc0on' “crank”' • Rocket'equa0on:'rela0ng'ΔV'and'change'in'mass'

m & minitial # initial ΔV = g I ln$ ! or m final = 0 SP $ m ! % final " & ΔV # exp$ ! % g0ISP " 2 'where'g0'='9.806'm/s ''' Isp'

• Isp'is'the'speciﬁc'impulse'(units'of'seconds)'and' measures'the'performance'of'the'rocket'' ΔV'

• ΔV'(delta(V)'represents'the'instantaneous'change'in' velocity'(from'current'velocity'to'the'desired'velocity).'

Circular Orbit Velocity ΔV = V2 − V1 µ V = C R

Elliptical Orbit Velocity

& 2 1 # VE = µ$ − ! % R a " Tangent'Burns'

• Ini0ally'you'are'in'a'circular'orbit' Your Initial Orbit with'radius'R1'around'Earth' µ V C1 = R1 R1

Your Spacecraft Tangent'Burns'

• Ini0ally'you'are'in'a'circular'orbit'

with'radius'R1'around'Earth' µ V C1 = R1 R1 ' R2 • You'perform'a'burn'now'which' puts'in'into'the'red(do^ed'orbit.'' So'you'perform'a'ΔV.'

V V V Δ = E1 − C1

ΔV Your new orbit after the ΔV burn Tangent'Burns'

• Ini0ally'you'are'in'a'circular'orbit'

with'radius'R1'around'Earth' µ V C1 = R1 R1 ' R2 • You'perform'a'burn'now'which' puts'in'into'the'red(do^ed'orbit.'' So'you'perform'a'ΔV.' V V V Δ = E1 − C1 where # 2 1 & 1 V and a R R E1 = µ% − ( = ( 1 + 2 ) $ R1 a ' 2 Hohmann'Transfer' The most efficient 2-burnmaneuver to transfer between 2 co-planar circular orbit

ΔV 2 ΔV = ΔV1 + ΔV2 µ & 2R # R1 V $ 2 1! Δ 1 = $ − ! R2 R1 % R1 + R2 " µ & 2R # V $1 1 ! Δ 2 = $ − ! R2 % R1 + R2 "

ΔV 1 1 a3 TOF = P = π 2 µ Hohmann'Transfer'(non(circular)'

• Another'way'to'view'it'is'using'angular'momentum'

! $ rarp h = 2µ# & " ra + rp %

ΔVtotal _3 = ΔVA + ΔVB

ΔVtotal _3' = ΔVA' + ΔVB' Hohmann'Transfer'(non(circular)'

! $ V h / r V h / r rArA' A 1 = 1 A A 3 = 3 A h1 = 2µ# & " rA + rA' % V h / r V h / r B 1 = 2 B B 3 = 3 B V = h / r V = h / r ! $ A' 1 1 A' A' 3' 3' A' rBrB' h2 = 2µ# & VB' = h2 / rB' VB' = h3' / rB' " rB + rB' % 1 3'

! $ ΔVA = abs VA −VA rArB ( 3 1) h3 = 2µ# & " rA + rB % ΔV = abs V −V A' ( A' 3 A' 1)

! $ ΔVB = abs VB −VB rA'rB' ( 2 3 ) h3' = 2µ# & " rA' + rB' % ΔV = abs V −V B' ( B' 2 B' 3' ) Bi(Ellip0c'Transfer' Bi(Ellip0c'Transfer'

If rc/ra < 11.94 then Hohmann is more efficient If rc/ra > 15.58 then Bi-Elliptic is more efficient Non(Hohmann'Transfers'w/ Common'Line'of'Aspides' Non(Hohmann'Transfers'w/ Common'Line'of'Aspides'

rA − rB etransfer = − rA cosθA − rB cosθB " % cosθA − cosθB htransfer = µrArB $ ' # rA cosθA − rB cosθB & A

2 2 ΔVA = V1 +Vtrans − 2V1Vtrans cosΔγ A

Δv vr@A − vr@A tanφ = r@A = orbit2 orbit1 v v v Δ ⊥@A ⊥@A orbit2 − ⊥@A orbit1 Non(Hohmann'Transfers'w/ Common'Line'of'Aspides'

2 2 ΔVA = V1 +Vtrans − 2V1Vtrans cosΔγ A

" µ / h e sinθ % Δγ = γ −γ γ = tan−1 ( 1) 1 1 A ( trans 1) @A 1 $ ' # h1 / rA & $ ' −1 Vr " % γi = tan & ) −1 (µ / htrans )etrans sinθ1 V γtrans = tan $ ' % ⊥ (i @A # htrans / rA & Apse'Line'Rota0on' Opportunity to transfer from one orbit to another using a single maneuver occurs at intersection points of the orbits

Apse Line Rotation Angle

η =θ1 −θ2

NOTE:

rorbit1@I = rorbit2@I h2 / µ h2 / µ 1 = 2 1+ e1 cosθ1 1+ e2 cosθ2 Apse'Line'Rota0on' Opportunity to transfer from one orbit to another using a single maneuver occurs at intersection points of the orbits

Setting θ2 =θ 1 −η and using the following identity

cos(θ1 −η) = cosθ1 cosη +sinθ1 sinη

Then θ1 has two solution corresponding to point I and J " 2 2 % −1 h1 − h2 θ1 = ζ ± cos $ 2 2 cosζ ' # e1h2 − e2h1 cosη & " 2 % −1 −e2h1 sinη ζ = tan $ 2 2 ' # e1h2 − e2h1 cosη & Apse'Line'Rota0on' Opportunity to transfer from one orbit to another using a single maneuver occurs at intersection points of the orbits

Given the orbit information of the two orbits

Next, compute r, v⊥i, vri, γi using θ1 and θ2 =θ1 −η

2 h / µ v⊥2 = h2 / r r = 1 1+ e1 cosθ1 vr2 = (µ / h2 )e2 sinθ2 −1 v⊥1 = h1 / r γ 2 = tan (vr2 / v⊥2 ) vr1 = (µ / h1)e1 sinθ1 2 2 v2 = vr2 + v⊥2 −1 γ1 = tan (vr1 / v⊥1)

2 2 2 2 v1 = vr1 + v⊥1 Δv = v1 + v2 − 2v1v2 cos(γ 2 −γ1) Apse'Line'Rota0on' Opportunity to transfer from one orbit to another using a single maneuver occurs at intersection points of the orbits

Δv vr@I − vr@I tanφ = r@I = orbit2 orbit1 v v v Δ ⊥@I ⊥@I orbit2 − ⊥@I orbit1

If you want to find the effect of a Δv on orbit 1 at θ1

v + Δv v + Δv v2 tanθ = ( ⊥1 ⊥ )( r1 r ) ⊥1 2 2 / r (v⊥1 + Δv⊥ ) e1 cosθ1 +(2v⊥1 + Δv⊥ )Δv⊥ (µ )

rv ⊥1 (at periapsis) If θ1 = vr = Δv⊥ = 0 then tanη = − Δvr µe1 Apse'Line'Rota0on' Opportunity to transfer from one orbit to another using a single maneuver occurs at intersection points of the orbits

Another useful equation:

2 (h1 + r1Δv⊥ ) e1 cosθ1 +(2h1 + r1Δv⊥ )r1Δv⊥ e2 = 2 h1 cosθ2 Plane'Change'Maneuvers' In general a plane change maneuver changes the orbit plane

Δv = v2 − v1 = (vr2 − vr1)uˆr + v⊥2uˆ⊥2 − v⊥1uˆ⊥1 Δv = Δv ⋅ Δv δ v2 2 2 2 Δv = (vr2 − vr1) + v⊥1 + v⊥1 − 2v⊥2v⊥1 cosδ v1 or

2 2 # % Δv = v1 + v2 − 2v1v2 $cosΔγ − cosγ1 cosγ 2 (1− cosδ)&

where Δγ = γ 2 −γ1 Plane'Change'Maneuvers' In general a plane change maneuver changes the orbit plane

If there is no plane change, δ = 0 then

2 2 No plane Δv = v + v − 2v v cosΔγ 1 2 1 2 change and we get the same equation as from slide 130.

If v r1 = vr2 = 0 a nd v⊥1 = v 1 a nd v⊥ 2 = v2 then

Pure plane 2 2 change Δv = v + v − 2v v cosδ 1 2 1 2 (common line of apisdes) Plane'Change'

"δ % " Δi % Pure plane change ΔvPC = 2vC sin$ ' = 2vC sin$ ' # 2 & # 2 & (circular to circular)

2 2 Δv(a) = (v2 − v1) + 4v1v2 sin (δ / 2)

Δv(b) = 2v1 sin(δ / 2) + v2 − v1 Δv(c) = v2 − v1 + 2v2 sin(δ / 2) Rendezvous' Catching a moving target: Hohmann transfer assume the target will be there independent of time, but for rendezvousing the target body is always moving.

target TIME = 0

n TOF ΔV φ0 = π − 2 ⋅ φ0 where µ n2 = 3 ≡ mean motion spacecraft a2 Rendezvous' Catching a moving target: Hohmann transfer assume the target will be there independent of time, but for rendezvousing the target body is always moving.

TIME = TOF

n TOF ΔV φ0 = π − target ⋅ φ0 target where µ ntarget = 3 ≡ mean motion spacecraft atarget Wait'Time'(WT)' How long do you wait before you can perform (or initiate) the transfer?

TIME = -WT

target φ0 φ -φ WT = 0 ntarget − nsc

ΔV

φ spacecraft Co(orbital'Rendezvous' Chasing your tail Forward thrust to slow Phasing orbit size down target 1/3 & 2 # , 2π −φinitial ) aphasing = $µ* ' ! φ0 $ * 2π ⋅n ' ! % + target ( " ΔV spacecraft

Reverse thrust to speed up Patched(Conic' An approximation breaks the interplanetary trajectory into regions where conic approximation is applicable

Sphere of Influence

2/5 & m # R a $ planet ! SOI = planet $ ! % mSun " Patched(Conic' An approximation breaks the interplanetary trajectory into regions where conic approximation is applicable • Computational Steps – PATCH 1: Compute the Hohmann transfer ΔVs – PATCH 2: Compute the Launch portion – PATCH 3: Compute the Arrival portion

• PATCH 1

– The ΔV1 from the interplanetary Hohmann is V∞ at Earth

– The ΔV2 from the interplanetary Hohmann is V∞ at target body arrival Patched(Conic' An approximation breaks the interplanetary trajectory into regions where conic approximation is applicable

RSOI Earth Departure V∞ @Earth (assume circular orbit) VEarth 2 & -V * # ΔV = V $ 2 + + ∞@departure ( −1! DEP CDEP + ( $ VC ! %$ , DEP ) "!

where µ V = DEP CDEP R CDEP RC@DEP ΔVDE P Patched(Conic' An approximation breaks the interplanetary trajectory into regions where conic approximation is applicable

ΔV CA Target Body Capture P (assume circular orbit)

2 & -V * # ΔV = V $ 2 + + ∞@arrival ( −1! CAP CCAP + ( $ VC ! %$ , CAP ) "! RC@CAP where RSOI µ V = CAP CCAP R CCAP

VMars V∞ @Mars Patched(Conic' An approximation breaks the interplanetary trajectory into regions where conic approximation is applicable

Total Mission ΔV for a simple 2 burn transfer

ΔVMISSION = ΔVDEP + ΔVCAP

ΔVDE P

ΔVCA P Planetary'Flyby' Getting a boost

• Gravity(assists'or'ﬂybys'are'used'to' – Reduce'or'increase'the'heliocentric'(wrt.'Sun)'energy'of'the'spacecra6' (orbit'pumping),'or' – Change'the'heliocentric'orbit'plane'of'the'spacecra6'(orbit'cranking),' or'both' Planetary'Flyby' Getting a boost Planetary'Flyby' Getting a boost Planetary'Flyby' Getting a boost Stationary Jupiter Moving Jupiter Planetary'Flyby' Getting a boost

Fg Fg

Vplane t Decrease energy wrt. Sun Increases energy wrt. Sun Planetary'Flyby' Getting a boost

ΔV • V∞'leveraging'is'the'use' of'deep'space'maneuvers' to'modify'the'V∞'at'a' body.''A'typical'example' post is'an'Earth'launch,'ΔV' maneuver If no maneuver'(usually'near' maneuver apoapsis),'followed'by'an' Earth'gravity'assist'(ΔV( EGA).'