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A Dilettante Dabbles in Orbital Mechanics

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A Dilettante Dabbles in Orbital Mechanics 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 r =R 3 p = 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 R=1 AU rp=0.21 Venus ☀ Venus Delta IV-Heavy: ∆ve = 12.7 km/s PSP New Horizons Earth R=0.72 AU Changing reference Earth PSP PSP ☀ Venus Venus Venus Earth R=1 AU PSP Earth ∆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.
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