What  do  we  need  for  a  mission  to  Mars?  

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Mo3ons  to  consider  

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1)  Orbital  mo3on  of  the  Earth  2)  Orbital  mo3on  of  Mars  3)  Launch  of  spacecra?  off  Earth  4)  Escape  of  spacecra?  from  Earth    5)  Orbital  mo-on  of  the  spacecra5  6)  Capture  of  spacecra?  by  Mars  

Hohmann  Transfer  

1.5  AU   1.0  AU  

Kepler’s  and  Newton’s  laws  provide  a  way  to  calculate  the  path  between  to  bodies  in  the  solar  system.  

What  is  the  semimajor  axis  of  this  orbit?  

2a  =  1.5  AU  +  1  AU  =  2.5  AU    

a  =  1.25  AU  

Hohmann  Transfer:    transfer  orbit  that  requires  the  minimum  energy  (usually)  

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Mars’  orbit  

Earth’s  orbit  

spacecra?’s  orbit  

Hohmann  Transfer  

1.5  AU   1.0  AU  

Kepler’s  and  Newton’s  laws  provide  a  way  to  calculate  the  path  between  to  bodies  in  the  solar  system.  

What  is  the  semimajor  axis  of  this  orbit?  

a  =  1.25  AU  

Hohmann  Transfer:    transfer  orbit  that  requires  the  minimum  energy  (usually)  

What  is  the  3me  required?  

Kepler’s  3rd  Law:  P2  =  a3  

P  =  (a3)1/2    P  =  (1.253)1/2    =  1.4  yrs  

Travel  3me  =  0.7  years  =  8.4  months   4  

Mars’  orbit  

Earth’s  orbit  

spacecra?’s  orbit  

Earth–Mars  (Hohmann)  Transfer  Orbit:  How  much  change  in  velocity  is  needed?  

For a circular orbit

Transfer orbit is actually elliptical so velocity depends on location in orbit (this results from conservation of energy and Kepler’s 2nd law regarding equal areas in equal times)

Pavorbitπ2

=

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1.5  AU   1.0  AU  

Mars’  orbit  

Earth’s  orbit  

spacecra?’s  orbit  

V1  V2  

Earth–Mars  Transfer  Orbit:  How  much  change  in  velocity  is  needed?  

•  We can calculate this.

•  Our satellite must leave going 0.8 km/sec faster than Earth and arrive at Mars going 2.4 km/sec slower than Mars.

V1 = 30.6 km/sec

V2 = 21.8 km/sec

•  Recall that the Earth and Mars are moving at 29.8 km/sec and 24.2 km/sec.

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1.5  AU   1.0  AU  

Mars’  orbit  

Earth’s  orbit  

spacecra?’s  orbit  

V1  V2  

Mo3ons  to  consider  

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1)  Orbital  mo3on  of  the  Earth  2)  Orbital  mo3on  of  Mars  3)  Orbital  mo3on  of  the  spacecra?  4)  Launch  of  spacecra?  off  Earth  5)  Escape  of  spacecra?  from  Earth    6)  Capture  of  spacecra5  by  Mars  

Will  Mars  capture  the  spacecra??  •  Spacecraft traveling

2.4 km/s slower than Mars’ orbital velocity

•  This is less than the escape velocity for Mars (5 km/s)

•  Hence, spacecraft is captured by Mars’ gravity when it arrives near the planet

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Mission  Plan:  How  long  would  a  round  trip  Earth-­‐Mars  mission  take?  

•  Period of transfer orbit is 1.4 years (from P2 = a3)

•  So Earth to Mars takes 0.7 years or 8.4 months

•  Planets are orbiting the sun, so we have to launch at just the right time for the spacecraft to rendezvous with Mars

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This  is  a  roundtrip  3cket  right?  •  Need to leave Mars

when Earth is 8.4 months behind the location where the transfer orbit will take the spacecraft

•  But when we arrive at Mars, Earth is only 3.6 months behind

•  We need to wait 15.4 months after arriving for Earth and Mars to line up right

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Mission  to  Mars!  •  Total mission time is then 8.4 + 15.4 +8.4 =32.2

months or 2.6 years

Courtesy  of  Touchstone  Pictures  

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What  could  be  some  of  the  hazards/problems  associated  with  a  2.6  year  journey?  

There  is  an  express  flight,  but  it  will  cost  you!  

•  There is more than 1 transfer orbit

•  After reaching escape velocity, accelerate spacecraft to 7 km/s instead of 0.8 km/s

•  Earth to Mars in 3 months

•  But higher velocity means higher fuel costs…because you have to slow down at Mars!

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13  What  about  hazards  for  this  journey?  

Return  travel  3me  =  4  months  

How  much  more  energy  needed  for  fast  mission?  

•  Kinetic energy is Ek = ½ mv2

•  If payload is mass is m=2000 kg, then slow mission: Ek = ½ (2000 kg) (0.8 km/s)2=6.4 x 108 Joules fast mission: Ek = ½ (2000 kg) (7 km/s)2=4.9 x1010 Joules

Fast mission to Mars requires 76 times more energy

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Inclined  Orbits  We  have  talked  about  orbits  and  implied  that  they  were  in  the  equatorial  plane.    These  orbits  are  called  equatorial  orbits.  

You  can  have  orbits  with  arbitrary  inclina3ons.  

If  the  angle  of  inclina3on  is  90°,  the  we  call  it  a  polar  orbit.    Why  use  a  polar  orbit?  

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Orbital  period  is  ~  100  minutes  Al3tude  ~  1000  km  

Crea3ve  Solu3on:  Molniya  Orbits  Good  orbit  design  can  make  other  problems  simpler.  

Russians  wanted  military  communica3ons  satellite  network  in  the  1960s.  

Geosta3onary  satellites  are  too  far  south  from  Russia  –  need  an  extremely  powerful  radio.  

Solu3on:  Molniya  orbit  1.  Highly  ellip3cal  2.  Inclined  orbit  63.4°  3.  Need  only  3  satellites  4.  Each  one  spends  about  8  hours  over  

Russia  

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Hyperbolic  &  Parabolic  Orbits  All  of  the  orbits  so  far  have  been  ellipses  (including  circular  orbits).  When  v  =  vescape    at  closest  approach,  we  call  this  a  parabolic  orbit.  

Types  of  Parabolic  Orbits:  1.  Escape  orbit  –  object  has  the  

escape  velocity  and  is  moving  away  from  the  planet  

2.  Capture  orbit  –  object  has  the  escape  velocity  and  is  moving  towards  the  planet  

Hyperbolic  orbits  have  a  speed  greater  than  the  escape  velocity  at  closest  approach.  

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Lagrange  Points  &  Halo  Orbits  Lagrange  Points:  a  small  body  under  the  influence  of  gravity  will  remain  stable  

L1  –  Gravity  of  two  bodies  is  in  balance  

L2  –  Gravity  of  two  bodies  balances  the  centrifugal  force  

L3  –  Slightly  inside  orbit;  affected  by  both  bodies  

L4  –  60°  in  front  of  orbi3ng  body  (distances  to  both  masses  are  equal)  

L5  –  60°  in  front  of  orbi3ng  body    Examples:  L1:  SOHO  L2:  WMAP  L4  &  L5:  Trojan  asteroids  at  Jupiter  

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Lagrange  Points  &  Halo  Orbits  

Halo  Orbit:  3D  orbit  near  L1,  L2,  or  L3    Complicated  structure  due  to  3-­‐Body  Problem.    Orbits  are  unstable  so  sta3on  keeping  is  required.    Examples:  SOHO,  Genesis  

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Orbital  Maneuvers  We  have  already  discussed  one  type  of  orbital  maneuver,  the  Hohmann  

transfer,  when  we  mapped  out  a  poten3al  mission  to  Mars.  

Hohmann  transfer:    1.  Transfer  between  two  

coplanar  circular  orbits  2.  Requires  two  engine  burns  3.  Lowest  energy  transfer  if    

R’/R  <  12  4.  Assume  impulsive  thrust  

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Orbital  Maneuvers  Other  orbital  maneuvers  are  required  for  different  applica3ons.  

Bi-­‐ellip@c  transfer:    1.  Transfer  between  two  

coplanar  circular  orbits  2.  Requires  three  engine  

burns  3.  Lower  energy  than  

Hohmann  transfer  if      R’/R  >  12  

4.  Assume  impulsive  thrust  5.  Longer  3me  than  Hohmann  

transfer  

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R  R’  

Gravita3onal  Assist  Some3mes  using  a  Hohmann  transfer  or  bi-­‐ellip3c  transfer  is  too  energy  intensive.  

Our  rocket  cannot  carry  that  much  fuel.  

Gravita3onal  Assist:  1.  Used  to  increase  speed,  

decrease  speed,  and  change  direc3on  

2.  Relies  on  using  the  rela3ve  mo3on  of  the  planet  and  spacecra?  

3.  Saves  fuel,  3me,  and  money  

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Gravita3onal  Assist  Some3mes  using  a  Hohmann  transfer  or  bi-­‐ellip3c  transfer  is  too  energy  intensive.  

Our  rocket  cannot  carry  that  much  fuel.  

How  it  works  (simplified):  1.  Spacecra?  moves  towards  planet  

with  speed  v  (rela3ve  to  Sun)  2.  Planet  moving  towards  spacecra?  

at  speed  u  (rela3ve  to  Sun)  3.  Spacecra?  moves  at  speed  u  +  v  

with  respect  to  planet’s  surface  (incoming)    

4.  Spacecra?  moves  at  speed  u  +  v  with  respect  to  planet’s  surface  (outgoing)  

5.  Spacecra?  moves  away  from  the  planet  with  speed  2u+v  (rela3ve  to  Sun)   Is  this  all  science  fic3on?  

Oberth  Effect:  gravita3onal  assist  with  thrusters  

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Gravita3onal  Assist  Notable  uses:  1. Mariner  10  2.  Voyager  I  &  II  3.  Galileo  4.  Ulysses  5.  Cassini  6. MESSANGER  

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Gravita3onal  Assist  

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Gravita3onal  Assist  

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Gravita3onal  Assist  

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Aerobraking  

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Aerobraking  

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But  is  it  really  that  simple…  •  Lunar  Mascons  (aka  mass  concentra3ons)  make  stable  low  lunar  obits  difficult  (impossible?)  to  find.  

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Konopliv  et  al,  Icarus  150,  1–18  (2001).    

But  is  it  really  that  simple…  •  So  why  not  just  fly  high  al3tude  orbits?  Because  the  Earth  gravita3onally  disturbs  high  al3tude,  circular  orbits.  

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•  But  there  might  be  highly          ellip3cal,  high  inclina3on          orbits  that  could  be  stable            for  about  100  years.  

Todd  Ely  and  Erica  Lieb,  Stable  Constella3ons  of  Frozen  Ellip3cal  Inclined  Lunar  Orbits,    Journal  of  the  Astronau3cal  Sciences,  vol.  53,  No.  3,  July-­‐Sept  2005,  pp.  301-­‐316  

Other  Orbital  Maneuvers  

•  Orbital  Inclina3on  Change  •  Phasing  •  Rendezvous  •  Docking  

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Orbital  Sta3on-­‐Keeping  

Orbital  Sta@on-­‐Keeping:    firing  thruster  to  keep  a  spacecra?  in  a  par3cular  orbit  

Any  real  orbit  will  change  with  3me  due  to  perturba3ons  from  other  bodies  in  the  solar  system.  

Typically  a  small  set  of  thrusters  are  used.    These  are  called  the  aDtude  control  system  (ACS).  

Sta3on-­‐keeping  is  cri3cal  for  satellites  that  must  be  oriented  in  a  certain  direc3on  to  communicate  with  Earth  (communica3on  satellites)  

Example:    satellite  is  orbit  around  the  Earth  is  perturbed  by  the  Sun,  Moon,  Jupiter…  

Now  this  process  is  automated  by  an  onboard  computer  that  collects  telemetry  and  makes  correc3ons.  

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