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How to get anywhere in the 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 : -Venus-Earth-Earth-

… say what!? eccentricity: S/C orbits 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 0.29 < r < 1 AU Venus Mars ☐ New Horizons: Earth  Jupiter  out passing & Ultima Thule B D ☐ Cassini: C Earth  Venus  Earth Earth A Earth  Venus  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) ] 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 are Good?

characteristic scattering angle

∆θ

∆v

surface escape planetary orbital 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 ()

• 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