Lesson 11: Orbital Transfers II
10/6/2016 Robin Wordsworth ES 160: Space Science and Engineering: Theory and Applica ons Objec ves
• Introduce concept of sphere of influence • Study the patched conics approach to interplanetary transfers • Understand the physical principle behind flyby / gravity assist trajectories Spheres of Influence
h p://watchingamerica.com/nrchandelsblad000078.shtml Spheres of Influence: Astronomical
h p://ccar.colorado.edu/asen5050/projects/projects_2012/roberts/roberts_proj.htm Spheres of Influence
• Approximate mo on of spacecra is due to most influen al celes al body only • Defini on of ‘most influen al’ is subtle – Force balance does not work! – C.f. Moon around Earth Spheres of Influence
• Need to imagine spacecra in orbit mass of planet around each body in turn – Compare with accelera on due to second body • Earth’s SOI is mass of Sun – ~145 rE (big) Sun-planet – 6x10-3 AU (small!) separa on Spheres of Influence
• Need to imagine spacecra in orbit around each body in turn – Compare with accelera on due to second body • Earth’s SOI is
– ~145 rE (big) – 6x10-3 AU (small!)
Prussing & Conway, Orbital Mechanics Patched Conics
• An engineer’s solu on to our old nemesis: the intractable n-body problem of Newtonian mechanics • Pretend a spacecra ’s orbit is always a solu on to the 2-body equa on (and hence a conic sec on) • What the solu on is depends only on which sphere of influence you are in! Patched Conics
circular orbit Patched Conics
SUN SOI
circular orbit
EARTH SOI Patched Conics
SUN SOI
ΔvA
circular orbit
EARTH SOI Patched Conics
SUN SOI
ΔvA hyperbolic escape trajectory
circular orbit
EARTH SOI Patched Conics
SUN SOI
ΔvA hyperbolic ??? escape trajectory
circular orbit
EARTH SOI Patched Conics
SUN SOI (~everywhere)
EARTH Patched Conics
SUN SOI (~everywhere)
EARTH Patched Conics
SUN SOI (~everywhere)
EARTH An Example: Earth-Venus transfer
• We are in a 200 km LEO and want to pass 500 km over the surface of Venus VENUS Steps: • Hyperbolic escape trajectory from Earth • Hohmann transfer to Venus • Hyperbolic flyby trajectory past Venus with 500 km periapse
EARTH An Example: Earth-Venus transfer
v 2 A =1 v 1+R 2 r VENUS r2 R r /r ⌘ 2 1
r1 • Calculate outbound Hohmann transfer as before • R = 1 AU / 0.723 AU = 1.383
• v∞,E = 2.49 km/s EARTH
v∞,E An Example: Earth-Venus transfer
SUN SOI
vp hyperbolic perigee v∞,E escape trajectory r Now use energy p conserva on to get perigee speed of hyperbolic trajectory
• vp = 11.29 km/s ΔvA is now assessed at perigee (it is the circular orbit speed increase 1 2 µE 1 2 EARTH SOI vp = v required to achieve 2 rp 2 1 hyperbolic trajectory from circular orbit) • vA = vp vc ΔvA = 3.5 km/s An Example: Earth-Venus transfer
SUN SOI
VENUS SOI
rp vp Exercise: given periapsis (pericytherion) rp = 500 km, v∞,V calculate ΔvB required for orbital injec on Gravity Assist Trajectories (Planetary Flybys)
• A vital part of many interplanetary missions • First successful planetary flyby: Mariner 2 to Venus (NASA, 1962) • First successful gravity assist: Mariner 10 to Mercury (NASA, 1974)
h p://nssdc.gsfc.nasa.gov/image/spacecra /mariner02.gif h p://nssdc.gsfc.nasa.gov/planetary/image/mariner10_labelled.jpg The Voyager Flybys
h ps://vimeo.com/69465942
h p://solarsystem.nasa.gov/basics/bsf4-1.php h p://voyager.jpl.nasa.gov/spacecra /goldenrec.html
Gravity Assist Trajectories (Planetary Flybys)
• First consider a sta onary planet v∞,B • Direc on of velocity at infinity changes, magnitude does not • Spacecra KE is unaltered
v∞,A Gravity Assist Trajectories (Planetary Flybys)
• Rest frame v∞,B vE • What v∞,B happens to v2 spacecra
speed a er vE the flyby? • Is energy s ll vE conserved? v v ∞,A ∞,A v1 Gravity Assist Trajectories (Planetary Flybys)
v∞,B Δv
v∞,A v∞,B TURN ANGLE
Δv v1 vE v2
v =2v sin[ /2] 1 v∞,A Gravity Assist Trajectories (Planetary Flybys)
h p://solarsystem.nasa.gov/basics/grav/primer.php Group Discussion How would we calculate the delta-v budget and launch system requirements to transport a 2 kg brick from Earth to the surface of Mars?