Lesson 11: Orbital Transfers II. 10/6/2016 Robin Wordsworth ES 160: Space Science and Engineering: Theory and ApplicaCons

Transcription

1 Lesson 11: Orbital Transfers II 10/6/2016 Robin Wordsworth ES 160: Space Science and Engineering: Theory and ApplicaCons

2 ObjecCves Introduce concept of sphere of influence Study the patched conics approach to interplanetary transfers Understand the physical principle behind flyby / gravity assist trajectories

3 hip://watchingamerica.com/nrchandelsblad shtml Spheres of Influence

4 Spheres of Influence: Astronomical hip://ccar.colorado.edu/asen5050/projects/projects_2012/roberts/roberts_proj.htm

5 Spheres of Influence Approximate mocon of spacecraq is due to most influen+al celescal body only DefiniCon of most influencal is subtle Force balance does not work! C.f. Moon around Earth

6 Spheres of Influence Need to imagine spacecraq in orbit around each body in turn Compare with acceleracon due to second body Earth s SOI is ~145 r E (big) 6x10-3 AU (small!) mass of planet mass of Sun Sun-planet separacon

7 Spheres of Influence Need to imagine spacecraq in orbit around each body in turn Compare with acceleracon due to second body Earth s SOI is ~145 r E (big) 6x10-3 AU (small!) Prussing & Conway, Orbital Mechanics

8 Patched Conics An engineer s solucon to our old nemesis: the intractable n-body problem of Newtonian mechanics Pretend a spacecraq s orbit is always a solucon to the 2-body equacon (and hence a conic seccon) What the solucon is depends only on which sphere of influence you are in!

9 Patched Conics circular orbit

10 Patched Conics SUN SOI circular orbit EARTH SOI

11 Patched Conics SUN SOI Δv A circular orbit EARTH SOI

12 Patched Conics SUN SOI Δv A hyperbolic escape trajectory circular orbit EARTH SOI

13 Patched Conics SUN SOI Δv A??? hyperbolic escape trajectory circular orbit EARTH SOI

14 Patched Conics SUN SOI (~everywhere) EARTH

15 Patched Conics SUN SOI (~everywhere) EARTH

16 Patched Conics SUN SOI (~everywhere) EARTH

17 An Example: Earth-Venus transfer We are in a 200 km LEO and want to pass 500 km over the surface of Venus Steps: Hyperbolic escape trajectory from Earth Hohmann transfer to Venus Hyperbolic flyby trajectory past Venus with 500 km periapse VENUS EARTH

18 An Example: Earth-Venus transfer v A v 2 =1 r 2 1+R VENUS r 2 R r 2 /r 1 r 1 Calculate outbound Hohmann transfer as before R = 1 AU / AU = v,e = 2.49 km/s EARTH v,e

19 An Example: Earth-Venus transfer SUN SOI v,e hyperbolic escape trajectory v p circular orbit r p perigee Now use energy conservacon to get perigee speed of hyperbolic trajectory v p = km/s Δv A is now assessed at perigee (it is the speed increase required to achieve hyperbolic trajectory from circular orbit) Δv A = 3.5 km/s EARTH SOI 1 2 v2 p v A = v p µ E r p = 1 2 v2 1 v c

20 An Example: Earth-Venus transfer SUN SOI VENUS SOI Exercise: given r p = 500 km, calculate Δv B required for orbital injeccon v p r p periapsis (pericytherion) v,v

21 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) hip://nssdc.gsfc.nasa.gov/image/spacecraq/mariner02.gif hip://nssdc.gsfc.nasa.gov/planetary/image/mariner10_labelled.jpg

22 The Voyager Flybys heps://vimeo.com/ hip://solarsystem.nasa.gov/basics/bsf4-1.php hip://voyager.jpl.nasa.gov/spacecraq/goldenrec.html

23 Gravity Assist Trajectories (Planetary Flybys) First consider a staconary planet DirecCon of velocity at infinity changes, magnitude does not SpacecraQ KE is unaltered v,b v,a

24 Gravity Assist Trajectories (Planetary Flybys) Rest frame What happens to spacecraq speed aqer the flyby? Is energy scll conserved? v,b v,a v E v E v,b v 2 v E v,a v 1

25 Gravity Assist Trajectories (Planetary Flybys) v,b Δv TURN ANGLE v,a v,b v E v 1 v 2 Δv v =2v 1 sin[ /2] v,a

26 Gravity Assist Trajectories (Planetary Flybys) hip://solarsystem.nasa.gov/basics/grav/primer.php

27 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?