How to get anywhere in the Solar System on a single tank of fuel
A dilettante dabbles in orbital mechanics
Dana Longcope Montana State University Parker Solar Voyager 1 & 2 Probe (PSP)
Gravity Assists: Cassini Why? How? Why so many? From Fran Bagenal’s lecture yesterday:
Sample Trajectory to Uranus: Earth-Venus-Earth-Earth-Jupiter Gravity Assist
… say what!? eccentricity: S/C orbits Sun ballistically:
ellipse w/ r p Sun @ focus
☀ (Kepler’s 1st) v
ra eccentricity:
rp
☀
R v
ra R r rp p ra ☀ v v
vo ra vo rp=0.04 R
☀ R
= 1.04 v a r R vo
vo rp=0.04 R
☀ =R
a ra=R r orbits grazing r=R v R vr=0 R rp=R = 3
a orbits r grazing r=R R r =R ☀ p v
vr=0 R scape orbit scape ☀ e v
vo R=1 AU Match the mission to the orbit ☐ MAVEN: Earth Mars direct Helios 2: Jupiter Mercury ☐ 1976: sampled solar wind 0.29 < r < 1 AU Venus Mars ☐ New Horizons: Earth Jupiter out passing Pluto & Ultima Thule B D ☐ Cassini: C Earth Venus Earth Earth A Earth Venus Saturn Earth Mars r a rp C ☐ MAVEN: ☀ Earth Mars direct
☐ Helios 2: Hohmann Transfer Orbit 1976: sampled solar wind 0.29 < r < 1 AU Mars ☐ New Horizons: Earth Jupiter out passing Pluto & Ultima Thule B D ☐ Cassini: C Earth Venus Earth A Earth Venus Saturn Mercury Earth r a C ☐ MAVEN: rp ☀ Earth Mars direct A ☐ Helios 2: 1976: sampled solar wind 0.29 < r < 1 AU
☐ New Horizons: Earth Jupiter out passing Pluto & Ultima Thule B D ☐ Cassini: C Earth Venus Earth Earth A Venus Saturn R=1 AU Match the mission to the orbit C ☐ MAVEN: Earth Mars direct A Helios 2: Jupiter Mercury ☐ 1976: sampled solar wind 0.29 < r < 1 AU Venus Mars D ☐ New Horizons: Earth Jupiter out passing Pluto & Ultima Thule B D ☐ Cassini: C Earth Venus Earth Earth A Earth Venus Saturn R=1 AU Match the mission to the orbit C ☐ MAVEN: Earth Mars direct A Helios 2: Jupiter Mercury ☐ 1976: sampled solar wind 0.29 < r < 1 AU Venus Mars D ☐ New Horizons: Earth Jupiter out passing Pluto & Ultima Thule
B B D ☐ Cassini: C Earth Venus Earth Earth A Venus Saturn Relative to Earth: Venus ) 1977 ( 2 2 New Horizons New PSP Voyager v
R=1 AU Helios 2 (1976) u + v m
u v
Impulse:
Specific [m/s] momentum/mass impulse: u + v m
u v depends on: 1. mass ratio fuel : payload 2. specific impulse u
Specific impulse depends on fuel:
2H2 + O2 2 × ( H2O + 2.5 eV ) Enthalpy of reaction per mass of reactants u 4 km/s
Fuel≃ required: Fuel required: u = 4 km/s 1. Trip out of solar system: ☀ ∆ve = 12 km/s
20
2. Trip to Sun (0.045 AU)
∆ve = 21 km/s Earth vf
vi 190 ∆v Earth Venus r p =0.21 R=1 ☀ AU PSP ∆v Delta IV e Venus = 12.7 Horizons New - Heavy: km/s Earth Venus reference Changing ☀ PSP R=0.72 R=1 Earth AU AU Earth PSP PSP Venus Venus ∆v for free: R=0.72 AU Venus gravity assist
☀
PSP
vi vf
vv vv Kepler meets Rutherford
☀ ’ ∆θ vf
Venusian F vi frame
vf ’ ’ vf vi elastic collision vv |v ’| = |v ’| f i ’ vi lose ang. mom. lose energy
∆θ=50ο ∆θ
vi’=23 km/s
vf’=23 km/s Rutherford scattering angle
’ vi = 23 km/s ∆θ v ’ ∆θ=50ο f p ve,p = 28 km/s p = 770 km surface of Venus @ v ’ Rv = 6,000 km i ’ vi = 23 km/s ∆θ ’ vf p > Rv = 6,000 km p ve,p < 10.0 km/s ∆θ < 9.7 ο
ο 50 > 5 ∆θ ’ vi 6+ flybys for rp = 0.046 AU rp = 0.046 AU 0.2 AU b ∆θ=9.7ο a elastic collision c ’ ’ |v | = |vi | a ☀ b h c h Venus 8 orbits of PSP Earth 7 flybys Why not just 6? Why the crazy spacing? Kepler 3rd: orbital period
orbit b: P= 2/3 P v resonance: 2 orbits by Venus P= (m/n) P 3 orbits by PSP v in 450 days m Venus orbits return to * = n PSP orbits d b
☀ b f * P = 3/7 1/2 2/3 1 Venus years vφ ☀
vv
v vf i outbound slingshot vr in-in or out-out rendezvous Psc m : n resonance Pv m Pv = n Psc ☀ in-out rendezvous: out in δv m Pv - δv = n Psc - δsc
δsc out-in rendezvous:
m Pv + δv = n Psc + δsc m:n = 2:3 Psc Pv 3:7 ☀ in-out 1:2 resonance out in out-in δv 2:5
δsc 2/5 3/7 1/2 2/3 1 orbits allowing repeated flybys
after ∆t ≤ 3 Pv ∆θ=9.7ο P largest jump: a sc ∆θ = o 9.1 b
☀ d in-out f resonance c out in e out-in δv g
δsc 2/5 3/7 1/2 2/3 1 legal in-out out-in [ via out-out (res) ] moves: out-in in-out [ via in-in (res) ] flyby 1 2 3 4 a ☀ b 5 c c h 6 e d b 7 d 7 g a h 675 225 450 res
441 197 in-out 3,4 239 days out-in 5,6 1,2 2226 days = 6 Earth years
destination: orbit b
ra= 0.94 AU rp= 0.16 AU ο ∆θ=9.7 a’ a a
☀ Venus b b
P = 2/3 Pv
1st orbit: a ∆θ=9.7ο 0.19 AU a’ & a’’ a ☀ b a’’
a a’
1st orbit: a launch R = 0.72 AU ∆θ=9.7ο Launch criteria:
• ∆ve < 12.8 km/s ’ • vi = 23 km/s • ∆θ < 9.7o
Launch ☀ window vr > 0
20 days ∆ve
vi’=23 km/s vr < 0
R = 1AU Center of the Launch window:
Venus lags ☀ o o Earth by 46 on: 46 35o 23 31 • 2015-06-05 51 days = 81o 35 km/s • 2017-01-09 • 2018-08-12 • 2020-03-20
v = 0 19 months apart r Which Planets are Good?
characteristic scattering angle
∆θ
∆v
surface escape planetary orbital speed speed Which Planets are Good?
characteristic scattering angle
Mercury Venus Earth Mars Jupiter
Vo [km/s] 48 35 30 24 13
Vesc,s [km/s] 4.2 10.4 11.2 5.0 60
Vesc,s / Vo 0.087 0.28 0.37 0.21 4.6 o o o o o ∆θch 0.4 4.5 7.2 2.4 132 Summary
• Launch speed (rel. to Earth) ∆ve limited by rocket • Can get extra ∆v from gravity assist: scattering off moving scatterer (planet)
• Elastic scattering |∆vp| conserved • Scattering angle = Rutherford scattering angle • Limited by surface escape speed • May need multiple scatterings (grav. assists) to achieve desired orbit • Multiple assists require resonant orbits, in-out or out-in rendezvous orbits