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Page 1: Orbital Mechanics Design Project

 

MAE  146:  Astronautics    Design  Project  Winter  2014  

   

Interplanetary  Orbiter  Mission    Neptune  Atmospheric  and  Interior  Science  Orbiter  

(NAISO)    

   

PREPARED  BY:    

Luke  Guirguis  Jose  Sepulveda  Sean  Godinez  

   

Page 2: Orbital Mechanics Design Project

 

CONSTANTS  and  KNOWN  VALUES    

Sun  (S):  

πΊπ‘Ÿπ‘Žπ‘£π‘–π‘‘π‘Žπ‘–π‘œπ‘›π‘Žπ‘™  π‘ƒπ‘Žπ‘Ÿπ‘Žπ‘šπ‘‘π‘’π‘Ÿ ∢  πœ‡  !"# =  132,712,000,000    π‘˜π‘š!

𝑠!  

Earth  (E):  

πΊπ‘Ÿπ‘Žπ‘£π‘–π‘‘π‘Žπ‘–π‘œπ‘›π‘Žπ‘™  π‘ƒπ‘Žπ‘Ÿπ‘Žπ‘šπ‘‘π‘’π‘Ÿ:    πœ‡  !  =  398,600  π‘˜π‘š!

𝑠!  

πΈπ‘Žπ‘Ÿπ‘‘β„Ž  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘…π‘Žπ‘‘π‘–π‘’𝑠  π‘Žπ‘Ÿπ‘œπ‘’𝑛𝑑  π‘†π‘’𝑛:  π‘…!  =  149.6 βˆ— 10!  π‘˜π‘š  

π‘…π‘Žπ‘‘π‘–π‘’π‘   π‘œπ‘“  πΈπ‘Žπ‘Ÿπ‘‘β„Ž:    π‘Ÿ!  =  6,378  π‘˜π‘š  

π‘ƒπ‘Žπ‘Ÿπ‘˜π‘–π‘›π‘”  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘…π‘Žπ‘‘π‘–π‘’𝑠  π‘Žπ‘‘  πΈπ‘Žπ‘Ÿπ‘‘β„Ž:    π‘Ÿ!"#$%&'#,! =  6,628  π‘˜π‘š  

π‘‡π‘Ÿπ‘’π‘’  π΄π‘›π‘Žπ‘œπ‘šπ‘™π‘¦  π‘œπ‘›  π‘€π‘Žπ‘Ÿπ‘β„Ž  13π‘‘β„Ž, 2014:      πœƒ!,! =  70.90868Β°    

π‘€π‘’π‘Žπ‘›  π‘€π‘œπ‘‘π‘–π‘œπ‘›  π‘œπ‘“  πΈπ‘Žπ‘Ÿπ‘‘β„Ž:  π‘›! =  1.1407 βˆ— 10!!𝑑𝑒𝑔𝑠  

π‘‰π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦  π‘œπ‘“  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  π‘…π‘’π‘™π‘Žπ‘‘𝑖𝑣𝑒  π‘‘π‘œ  π‘†π‘’𝑛:  π‘£! =  29.7844π‘˜π‘šπ‘       

Neptune  (N):  

πΊπ‘Ÿπ‘Žπ‘£π‘–π‘‘π‘Žπ‘–π‘œπ‘›π‘Žπ‘™  π‘ƒπ‘Žπ‘Ÿπ‘Žπ‘šπ‘‘π‘’π‘Ÿ:    πœ‡  !  =  6,835,100  π‘˜π‘š!

𝑠!  

𝑁𝑒𝑝𝑑𝑒𝑛𝑒  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘…π‘Žπ‘‘π‘–π‘’𝑠  π‘Žπ‘Ÿπ‘œπ‘’𝑛𝑑  π‘†π‘’𝑛:    π‘…!  =  4.495 βˆ— 10!  π‘˜π‘š  

π‘…π‘Žπ‘‘π‘–π‘’π‘   π‘œπ‘“  π‘π‘’𝑝𝑑𝑒𝑛𝑒:  π‘Ÿ!  =  24,760  π‘˜π‘š  

π‘‡π‘Žπ‘Ÿπ‘”π‘’π‘‘  πΆπ‘–π‘Ÿπ‘π‘’π‘™π‘Žπ‘Ÿ  π‘ƒπ‘Žπ‘Ÿπ‘˜π‘–𝑛𝑔  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘…π‘Žπ‘‘π‘–π‘’𝑠  π΄π‘Ÿπ‘œπ‘’𝑛𝑑  π‘π‘’𝑑𝑝𝑒𝑛𝑒 ∢    π‘Ÿ!"#$%! = 25,060    π‘˜π‘š        

π‘‡π‘Ÿπ‘’π‘’  π΄π‘›π‘Žπ‘œπ‘šπ‘™π‘¦  π‘œπ‘›  π‘€π‘Žπ‘Ÿπ‘β„Ž  13π‘‘β„Ž, 2014:      πœƒ!,! = 274.44241Β°  

π‘€π‘’π‘Žπ‘›  π‘€π‘œπ‘‘π‘–π‘œπ‘›  π‘œπ‘“  π‘π‘’𝑑𝑝𝑒𝑛𝑒:  π‘›! =  6.9260 βˆ— 10!!𝑑𝑒𝑔𝑠

 

π‘‰π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦  π‘œπ‘“  π‘π‘’𝑝𝑑𝑒𝑛𝑒  π‘…π‘’π‘™π‘Žπ‘‘𝑖𝑣𝑒  π‘‘π‘œ  π‘†π‘’𝑛:  π‘£! = 5.4336π‘˜π‘šπ‘   

Centaur  Upper  Stage:  

𝐼!" = 451  π‘   

πΆπ‘’π‘›π‘‘π‘Žπ‘’π‘Ÿ  π‘€π‘Žπ‘ π‘  = 2,243  π‘˜π‘”  

NASIO:  

𝐼!" = 318  π‘   

𝑁𝐴𝑆𝐼𝑂  π‘€π‘Žπ‘ π‘  = 1,800  π‘˜π‘”  

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VARIABLES    

Transfer  Phase  π‘Ž!:      π‘†π‘’π‘šπ‘– βˆ’π‘€π‘Žπ‘—π‘œπ‘Ÿ  π΄π‘₯𝑖𝑠  π‘œπ‘“πΈπ‘™π‘™π‘–π‘π‘‘π‘–π‘π‘Žπ‘™  π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘’π‘Ÿ  π‘‚π‘Ÿπ‘π‘–π‘‘    

𝑣!"! !:      π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  β„Žπ‘’π‘™π‘–π‘œπ‘π‘’π‘›π‘‘π‘Ÿπ‘–π‘  π‘£π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘Žπ‘‘  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  π‘‘π‘’π‘π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’  π‘“π‘œπ‘Ÿ  π‘Ž  π»π‘œβ„Žπ‘šπ‘Žπ‘›π‘›    π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘“π‘’π‘Ÿ          

𝑣!,!:      π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  π‘£π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘“π‘œπ‘Ÿ  π‘‘β„Žπ‘’  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  π‘π‘’π‘›π‘‘π‘Ÿπ‘–π‘  π‘‘π‘’π‘π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘’π‘ π‘π‘Žπ‘π‘’  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦  

𝑣!"#$%&'#,!:π‘‰π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦  π‘œπ‘“  πΆπ‘–π‘Ÿπ‘’π‘™π‘Žπ‘Ÿ  π‘ƒπ‘Žπ‘Ÿπ‘˜π‘–𝑛𝑔  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘Žπ‘Ÿπ‘œπ‘’𝑛𝑑  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  

𝑣!!",!: πΌπ‘›π‘–π‘‘π‘–π‘Žπ‘™  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘£π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘€π‘–π‘‘β„Žπ‘–π‘›  π‘ π‘β„Žπ‘’π‘Ÿπ‘’  π‘œπ‘“  π‘–𝑛𝑓𝑙𝑒𝑒𝑛𝑐𝑒  π‘œπ‘“  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  

Δ𝑣!:  π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  βˆ†π‘£  π‘‘π‘œ  π‘–π‘›π‘ π‘’π‘Ÿπ‘‘  π‘‘β„Žπ‘’  π‘ π‘π‘Žπ‘π‘’π‘π‘Ÿπ‘Žπ‘“π‘‘  π‘œπ‘›π‘‘π‘œ  π‘Ž  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  π‘’π‘ π‘π‘Žπ‘π‘’  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦  

𝑇!:      π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘’π‘Ÿ  π‘‡π‘–π‘šπ‘’  π‘œπ‘“  π‘†π‘π‘Žπ‘π‘’π‘π‘Ÿπ‘Žπ‘“π‘‘  

𝑇!:    π‘ƒπ‘’π‘Ÿπ‘–π‘œπ‘‘  π‘œπ‘“  π‘π‘’𝑑𝑝𝑒𝑛𝑒  π‘‚π‘Ÿπ‘π‘–π‘‘  

πœ™:      π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  π΄π‘›π‘”π‘’π‘™π‘Žπ‘Ÿ  π‘†π‘’π‘π‘Žπ‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›  π‘œπ‘“  π‘ƒπ‘™π‘Žπ‘›π‘’𝑑𝑠  π‘‘π‘œ  π‘…π‘’π‘›π‘‘π‘’π‘§π‘£π‘œπ‘’𝑠  π‘Žπ‘‘  π‘Žπ‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™  

𝛽!:      πΏπ‘œπ‘π‘Žπ‘‘π‘–π‘œπ‘›  π‘œπ‘“  π‘‘β„Žπ‘’  π‘π‘’π‘Ÿπ‘–π‘”π‘’π‘’  π‘œπ‘“  π‘‘β„Žπ‘’  π‘‘π‘’π‘π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘Ž  

βˆ†!:        π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  π‘Žπ‘–π‘šπ‘–π‘›π‘”  π‘‘π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’  π‘œπ‘“  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘’π‘ π‘π‘Žπ‘π‘’  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦    

Arrival  Phase  π‘£!"! !:      π»π‘’π‘™π‘–π‘œπ‘π‘’π‘›π‘‘π‘Ÿπ‘–π‘  π‘£π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘Žπ‘‘  π‘Žπ‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™    

𝑣!,!:      π‘‰π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘“π‘œπ‘Ÿ  π‘‘β„Žπ‘’  π‘π‘™π‘Žπ‘›π‘’π‘‘π‘œπ‘π‘’π‘›π‘‘π‘Ÿπ‘–π‘  π‘Žπ‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦  

𝛽!:      πΏπ‘œπ‘π‘Žπ‘‘π‘–π‘œπ‘›  π‘œπ‘“  π‘‘β„Žπ‘’  π‘π‘’π‘Ÿπ‘–π‘”π‘’π‘’  π‘œπ‘“  π‘‘β„Žπ‘’  π΄π‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘Ž  

βˆ†!:        π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  π‘Žπ‘–π‘šπ‘–π‘›π‘”  π‘‘π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’  π‘‘β„Žπ‘’  π‘ π‘π‘Žπ‘π‘’π‘π‘Ÿπ‘Žπ‘“π‘‘  π‘ β„Žπ‘œπ‘’𝑙𝑑  π‘’π‘›π‘π‘œπ‘’π‘›π‘‘π‘’π‘Ÿ  π‘‘β„Žπ‘’  π‘ π‘β„Žπ‘’π‘Ÿπ‘’  π‘œπ‘“  π‘–𝑛𝑓𝑙𝑒𝑒𝑛𝑐𝑒    

β„Ž!!",!:      π΄π‘›π‘”π‘’π‘™π‘Žπ‘Ÿ  π‘šπ‘œπ‘šπ‘’π‘›π‘‘π‘’π‘š  π‘œπ‘“  π‘‘β„Žπ‘’  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘Žπ‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦  

𝑒!!",! ∢      πΈπ‘π‘π‘’π‘›π‘‘π‘Ÿπ‘–π‘π‘–π‘‘π‘¦  π‘œπ‘“  π‘‘β„Žπ‘’  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘Žπ‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦  

𝑣!:      π‘ƒπ‘’π‘Ÿπ‘–π‘Žπ‘π‘ π‘–π‘   π‘£π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘œπ‘“  π‘‘β„Žπ‘’  β„Žπ‘¦π‘π‘’π‘Ÿπ‘π‘œπ‘™π‘–π‘  π‘‘π‘Ÿπ‘Žπ‘—π‘’π‘π‘‘π‘œπ‘Ÿπ‘¦    

Δ𝑣!:      π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘  βˆ†π‘£  π‘‘π‘œ  π‘–π‘›π‘ π‘’π‘Ÿπ‘‘  π‘–π‘›π‘‘π‘œ  πΆπ‘–π‘Ÿπ‘π‘’π‘™π‘Žπ‘Ÿ  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘Žπ‘Ÿπ‘œπ‘’𝑛𝑑  π‘π‘’𝑝𝑑𝑒𝑛𝑒  

𝑣!"#$%&'#,!:π‘‰π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦  π‘œπ‘“  πΆπ‘–π‘Ÿπ‘’π‘™π‘Žπ‘Ÿ  π‘ƒπ‘Žπ‘Ÿπ‘˜π‘–𝑛𝑔  π‘‡π‘Žπ‘Ÿπ‘”𝑒𝑑  π‘‚π‘Ÿπ‘π‘–π‘‘  π‘Žπ‘Ÿπ‘œπ‘’𝑛𝑑  π‘π‘’𝑝𝑑𝑒𝑛𝑒  

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CONTENTS    Section                     page        1.0        Mission  Objectives…………………………………………………          1        2.0        Launch  to  Parking  Orbit  Phase………………………............          2        3.0                                  Interplanetary  Transfer  Phase………………………………..          4        4.0                                  Planetary  Arrival  Phase………………………………………….          7        5.0                                  Propellant  Mass  Requirements……………………………….          9        6.0                                Mission  Summary…………………………………………………...        10        7.0              Team  Contributions…………………………………...................          12        8.0        References…………………………………………………………….          13

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1  

1.0 MISSION  OBJECTIVE    

The  Neptune  Atmospheric  and  Interior  Science  Orbiter  (NAISO)  will  be  orbiting  

Neptune  to  perform  an  in  depth  study  of  the  planet.  The  only  other  spacecraft  that  

has  flown  by  Neptune  is  Voyager  2,  which  revealed  Neptune  had  four  rings,  five  

moons,  and  a  "Great  Dark  Spot".  The  β€œGreat  Dark  Spot”  had  vanished  by  the  time  the  

Hubble  Space  Telescope  studied  Neptune  five  years  later.  NAISO  will  be  launched  

and  inserted  into  a  circular  parking  orbit  around  Earth,  sent  to  Neptune  using  a  

Hohmann  transfer,  and  inserted  into  a  circular  orbit  around  Neptune.  NAISO’s  

science  objectives  are:  

β€’ Study  Neptune’s  atmosphere  and  measure  composition,  temperature,  and  

other  atmospheric  properties  

β€’ Asses  what  the  β€œGreat  Dark  Spot”  was  and  possibly  search  for  other  

occurrences  

β€’ Map  Neptune’s  gravity  and  magnetic  fields  to  offer  insight  on  its  internal  

structure  and  core  

NAISO  is  equipped  with  various  instruments  similar  to  the  JUNO  spacecraft  sent  to  

Jupiter  operated  by  JPL,  in  order  to  carry  out  these  scientific  objectives.      

 

 

 

 

 

 

 

 

 

 

 

 

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2.0 LAUNCH  TO  PARKING  ORBIT  PHASE  

 

The  launch  of  mission  NAISO  is  scheduled  to  take  place  on  June  13th,  2014    

(calculations  shown  in  Section  3:  Interplanetary  Transfer  Phase)  under  the  

assumptions  that  the  orbits  of  the  planets  around  the  Sun  are  coplanar  and  

circular.  On  this  date,  the  required  angular  separation  of  Earth  from  Neptune  is  

approximately  113.1571Β°.  While  under  realistic  conditions  a  launch  would  not  

be  scheduled  until  June  16th,  2014.  The  launch  site  is  located  at  Cape  Canaveral,  

Florida,  also  known  as  the  Kennedy  Space  Center,  at  28.Β°  30'  N  Latitude  and  80Β°  

33'  W  Longitude.  The  vehicle  will  be  placed  in  parking  orbit  at  a  height  altitude  

of  250km  with  a  28.5Β°  inclination,  which  means  we  will  launch  with  an  Azimuth  

of  90Β°  because  of  the  equation:  

πΌπ‘›π‘π‘™π‘–π‘›π‘‘π‘Žπ‘–π‘œπ‘›:    π‘– = 28.5Β°            

πΏπ‘Ž = 28.5Β°  

𝐴𝑧 = sin!!cos 𝑖cos πΏπ‘Ž = 90Β°  

This  launch  Azimuth  will  allow  us  to  have  a  launch  window  of  around  2  hours  

according  to  the  graph  below:  

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According  to    NASA's  Horizon  web  interface,  on  the  date  of  departure  the  Sun  

relative  to  the  geocentric  frame  will  give  us  a  hyperbolic  asymptote  with  a  right  

ascension  of:    

π‘…π‘–π‘”β„Žπ‘‘  π΄π‘ π‘π‘’π‘›π‘–π‘œπ‘›:    π‘…𝐴 = 05: 24: 0.09  HMS    

and  a  declination  of  π·πΈπΆ = 23: 10: 46.7  DMS  

 

The  location  of  the  hyperbolic  asymptote  is  :  

𝛽! = 72.1256Β°  ,      π‘π‘Žπ‘™π‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›π‘   π‘–𝑛  π‘†π‘’π‘π‘‘π‘–π‘œπ‘›  03  

and  

βˆ†!= 9101.366  π‘˜π‘š,      π‘π‘Žπ‘™π‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›π‘   π‘–𝑛  π‘†π‘’π‘π‘‘π‘–π‘œπ‘›  03    

 

 

 

 

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3.0 INTERPLANETARY  TRANSFER  PHASE  

Once  in  a  circular  parking  orbit  around  Earth  (π‘Ÿ!"#$%&'#,! =  6,628  π‘˜π‘š),  the  

NAISO  satellite  will  need  to  attain  a  βˆ†v1  with  the  upper  stage  Centaur  V  to  begin  its  

Hohmann  transfer  to  Neptune.  In  order  to  calculate  this,  we  need  to  find  the  

required  heliocentric  velocity  of  the  spacecraft:  

 

π‘†π‘’π‘šπ‘– βˆ’π‘€π‘Žπ‘—π‘œπ‘Ÿ  π΄π‘₯𝑖𝑠  π‘œπ‘“  πΈπ‘™π‘™π‘–π‘π‘‘π‘–π‘π‘Žπ‘™  π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘’π‘Ÿ  π‘‚π‘Ÿπ‘π‘–π‘‘:  

π‘Ž! =12   𝑅! + 𝑅! =  2.3223e09  π‘˜π‘š  

 

π»π‘’π‘™π‘–π‘œπ‘π‘’π‘›π‘‘π‘Ÿπ‘–π‘  π‘£π‘’π‘™π‘œπ‘π‘–π‘‘𝑦  π‘Žπ‘‘  πΈπ‘Žπ‘Ÿπ‘‘β„Ž  π‘‘π‘’π‘π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’:  

𝑣!"! ! =  2πœ‡  !"#𝑅!

βˆ’πœ‡  !!"π‘Ž!

= 41.4376π‘˜π‘šπ‘   

Thus,  the  velocity  for  the  hyperbolic  departure  from  Earth  is:  

𝑣!,! = 𝑣!"! ! βˆ’ 𝑣! = 11.6532π‘˜π‘šπ‘   

The  location  of  the  perigee  of  the  departure  hyperbola  is  found  using  the  equations  

below:  

𝛽! = cos!!1

1+π‘Ÿ!"#$%&'#,!𝑣!,!!

πœ‡  !

  =  72.1256Β°    

 

βˆ†!= π‘Ÿ!"#$%&'#,! 1+2πœ‡  !

π‘Ÿ!"#$%&'#,! βˆ— 𝑣!,!!= 9101.366  π‘˜π‘š    

 

The  initial  hyperbolic  velocity    (  π‘£!!",!)  at  Earth  departure  is  found  using  

conservation  of  energy,  where  π‘Ÿ!"#$%&'#,!  is  the  radius  of  the  parking  orbit  :  

𝑣!!",!!

2 βˆ’πœ‡  !

π‘Ÿ!"#$%&'#,!=𝑣!,!!

2 βˆ’πœ‡  !"#π‘Ÿ!

 

 

Solving  for    π‘£!!",!  ,  the  equation  reduces  to:    

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 π‘£!!",! =   𝑣!,!! +2πœ‡  !

π‘Ÿ!"#$%&'#,!=  16.0023

π‘˜π‘šπ‘     

And  the  required  βˆ†v  to  attain  π‘£!,!  is:    

Δ𝑣! =  π‘£!!",! βˆ’ 𝑣!"#$%&'#,!        ,    where  π‘£!"#$%&'#,! =!  !

!!"#$%&'#,!=      

   Ξ”𝑣! =  π‘£!!",! βˆ’  !  !

!!"#$%&'#,!= 8.2474 !"

!  

 

The  required  angular  separation  (πœ™)  between  the  Earth  and  Neptune  at  the  time  of  

Earth  departure  in  order  to  arrive  at  Neptune’s  orbit  and  rendezvous  is  found  using  

the  time  of  the  Hohmann  transfer,  which  is  half  of  the  transfer  period  of  the  elliptical  

orbit:  

𝑇! =12𝑇 =

12

2πœ‹πœ‡  !"#

π‘Ž!!/! =  965099874  π‘     

 

One  period  of  Neptune’s  orbit  is:  

𝑇! =    !!!  !"#

𝑅!!/! =  5.1978𝑒09  s  

 

 

 Thus  the  required  angular  separation  is  found  using  the  relation:  180Β°βˆ’ πœ™180Β° =  

𝑇!(  π‘‡!/2)

       

 

solving  for  πœ™  yields:      πœ™ = 180Β°βˆ’ !!βˆ—!"#Β°  !!!

=  113.1571Β°  

 

 

 

 

 

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With  the  angular  separation,  we  can  now  calculate  the  launch  date.  Using  the  true  

anomalies  of  Earth  and  Neptune  recorded  on  March  13th,  2014  

(πœƒ!,! = 70.90868Β°    π‘Žπ‘›π‘‘  πœƒ!,! = 274.44241Β°),  and  also  the  mean  motion  of  each  

planet  (𝑛! =  βˆ— 10!!!"#!        π‘Žπ‘›π‘‘        π‘›! =  βˆ— 10!!

!"#!),  we  can  define  the  positions  of  

Earth  and  Neptune  relative  to  these  true  anomalies  with  the  equations:  

πœƒ! = πœƒ!,! +  π‘›!𝑑                π‘Žπ‘›π‘‘              πœƒ! = πœƒ!,! +  π‘›!𝑑  

The  separation  angle  is  the  difference  of  these  two  equations:  

πœ™ =  πœƒ! βˆ’ πœƒ! =  πœƒ!,! βˆ’  πœƒ!,!  +   𝑛! βˆ’ 𝑛! 𝑑  

Solving  for  time  t,  this  will  give  us  the  time  from  the  true  anomaly  date  in  seconds  to  

launch:  

𝑑 =πœ™ + πœƒ!,! βˆ’ πœƒ!,!

𝑛! βˆ’ 𝑛!=  7971148  π‘   

Using  the  MATLAB  function  β€œaddtodate(  )”,  this  time  t  was  added  to  the  initial  date  

of    March  13th,  2014  to  yield  a  launching  date  of  June  13th,  2014.  

 

The  arrival  date  is  simply  calculated  by  adding  the  transfer  time  (𝑇!)  to  the  launch  

date.  Using  MATLAB  again,  the  arrival  date  is  calculated  to  be  January  11th,  2045.  

 

 

 

 

 

 

 

 

 

 

 

 

 

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4.0 PLANETARY  ARRIVAL  PHASE  

 

Upon  arrival  to  Neptune,  the  heliocentric  velocity  of  the  spacecraft  is:  

𝑣!"! ! =  2πœ‡  !"#𝑅!

βˆ’πœ‡  !"#π‘Ž!

=  1.3791π‘˜π‘šπ‘   

Thus,  the  velocity  for  the  hyperbolic  arrival  to  Neptune  is:  

𝑣!,! = 𝑣!"! ! βˆ’ 𝑣! = βˆ’4.0545π‘˜π‘šπ‘   

The  minus  sign  indicates  that  Neptune  is  travelling  faster  than  the  orbiter  and  

catching  up  to  it,  and  thus  π‘£!,!  is  pointed  opposite  of  Neptune’s  velocity.  

 

The  target  orbit  radius  around  Neptune  (π‘Ÿ!"#$%! = 25,060    π‘˜π‘š)  is  300  km  above  the  

surface.  The  location  of  the  perigee  of  the  departure  hyperbola  is  found  using  the  

equation  below:  

𝛽! = cos!!1

1+π‘Ÿ!"#$%&'#,!𝑣!,!!

πœ‡  !

  = 19.4118  Β°    

 

The  required  aiming  distance  (βˆ†)  with  which  the  spacecraft  should  encounter  the  

sphere  of  influence  of  Neptune  in  order  achieve  the  target  orbit  radius  is  found  

using  the  equation:  

βˆ†!=  π‘Ÿ!"#$%! 1+2πœ‡  !

π‘Ÿ!"#$%! βˆ— 𝑣!,!!= 1.4652𝑒05  π‘˜π‘š    

This  provides  a  hyperbolic  arrival  trajectory  with  an  eccentricity  of  :  

𝑒!!",! =π‘Ÿ!"#$%! βˆ— 𝑣!,!!

πœ‡  !+ 1 = 1.0603  

and  a  specific  angular  momentum  of:  

β„Ž!!",! = π‘Ÿ!"#$%! 𝑣!,!! +2πœ‡  !π‘Ÿ!"#$%!

= 5.9405𝑒05π‘˜π‘š!

𝑠  

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Thus,  the  periapsis  velocity  of  the  hyperbolic  arrival  is:  

𝑣! =β„Ž!!",!π‘Ÿ!"#$%!

= 23.7052π‘˜π‘šπ‘   

The  velocity  required  for  the  circular  target  orbit  around  Neptune  is:  

𝑣!"#$%&'#,! =  πœ‡  !

π‘Ÿ!"#$%!= 16.5151

π‘˜π‘šπ‘   

Which  means  the  required  βˆ†v  to  circularize  the  orbiter’s  path  is:  

Δ𝑣! = 𝑣!"#$%&'#,! βˆ’ 𝑣! = βˆ’7.1901π‘˜π‘šπ‘     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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5.0 PROPELLANT  MASS  REQUIREMENTS  

The  NAISO  satellite  is  estimated  to  have  a  mass  that  is  half  of  the  JUNO  

spacecraft.  In  addition,  the  NAISO  is  using    

𝑁𝐴𝐼𝑆𝑂  πΌ!" = 318  π‘   

𝑁𝐴𝑆𝐼𝑂  π‘€π‘Žπ‘ π‘ :    π‘š!" = 1800  π‘˜π‘”  

Thus,  the  required  mass  of  propellant  (π‘š!,!)  needed  to  achieve  Ξ”𝑣! = 7.1901 !"!    is  

found  with  the  equation:  

π‘š!,!

π‘š!" +π‘š!,!= 1βˆ’ 𝑒

!( !!!!!"βˆ—!!

)  

Solving  for  π‘š!,!  yields:    

π‘š!,! = 16,240.48  π‘˜π‘”  

which  gives  a  total  mass  of  the  spacecraft  with  propellant  to  be:  

π‘š!"!! = π‘š!" +π‘š!,! =  18040.48  π‘˜π‘”  

 

Now  the  required  amount  of  fuel  for  Ξ”𝑣! =  8.2474 !"!  can  be  calculated:  

πΆπ‘’π‘›π‘‘π‘Žπ‘’π‘Ÿ  πΌ!" = 451  π‘   

πΆπ‘’π‘›π‘‘π‘Žπ‘’π‘Ÿ  π‘€π‘Žπ‘ π‘ :    π‘š!"#$%&' = 1800  π‘˜π‘”  

 

π‘š!,!

π‘š!"#$%&' +π‘š!,! +π‘š!"!!= 1βˆ’ 𝑒

!( !!!!!"βˆ—!!

)  

Again,  solving  for  π‘š!,!  yields:    

π‘š!,! = 110,549.34  π‘˜π‘”  

This  yields  a  total  mass  of  the  upper  stage  prior  to  the  execution  of  the  βˆ†v1  of:  

π‘š!"!#$ = π‘š!"#$%&' +π‘š!,! +π‘š!"!! = 130,832.82  π‘˜π‘”    

 

 

 

 

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6.0 MISSION  SUMMARY  

Objectives:  

β€’ Insert  Satellite  into  circular  orbit  300  km  above  Neptune’s  surface  with  a  

Hohmann  transfer  

β€’ Study  Neptune’s  atmosphere  and  measure  composition,  temperature,  and  

other  atmospheric  properties  

β€’ Asses  what  the  β€œGreat  Dark  Spot”  was  and  possibly  search  for  other  

occurrences  

β€’ Map  Neptune’s  gravity  and  magnetic  fields  to  offer  insight  on  its  internal  

structure  and  core  

 

Important  Launching  Parameters:  

               Launch  Date:  June  13th,  2014  

               Arrival  Date:  January  11th,  2045  

πΌπ‘›π‘π‘™π‘–π‘›π‘‘π‘Žπ‘–π‘œπ‘›:    π‘– = 28.5Β°            

πΏπ‘Žπ‘’π‘›β„Ž  πΏπ‘Žπ‘‘𝑖𝑑𝑒𝑑𝑒:  πΏπ‘Ž = 28.5Β°  

π΄π‘§π‘–π‘šπ‘’π‘‘β„Ž ∢ 𝐴𝑧 = 90Β°  

 

Important  Transfer  Parameters:  

   π‘£!,! = 11.6532π‘˜π‘šπ‘   

𝛽! = 72.1256Β°    

βˆ†!= 9101.366  π‘˜π‘š    

Δ𝑣! =  8.2474π‘˜π‘šπ‘   

πœ™ =  113.1571Β°  

 

Important  Arrival  Parameters:  

𝑣!,! = βˆ’4.0545π‘˜π‘šπ‘   

π‘Ÿ!"#$%! = 25,060    π‘˜π‘š  

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𝛽! = 19.4118  Β°    

 

βˆ†!=  1.4652𝑒05  π‘˜π‘š    

𝑣!"#$%&'#,! =  16.5151π‘˜π‘šπ‘   

Δ𝑣! = βˆ’7.1901π‘˜π‘šπ‘     

 

Mass  Budget:  

𝑁𝐴𝐼𝑆𝑂  πΌ!" = 318  π‘   

𝑁𝐴𝑆𝐼𝑂  π‘€π‘Žπ‘ π‘ :    π‘š!" = 1800  π‘˜π‘”  

π‘š!,! = 16,240.48  π‘˜π‘”  

π‘š!"!! =  18040.48  π‘˜π‘”  

 

πΆπ‘’π‘›π‘‘π‘Žπ‘’π‘Ÿ  πΌ!" = 451  π‘   

πΆπ‘’π‘›π‘‘π‘Žπ‘’π‘Ÿ  π‘€π‘Žπ‘ π‘ :    π‘š!"#$%&' = 1800  π‘˜π‘”  

π‘š!,! = 110,549.34  π‘˜π‘”  

 

 

π‘š!"!#$ = 130,832.82  π‘˜π‘”    

 

 

 

 

 

 

 

 

 

 

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7.0 TEAM  CONTRIBUTIONS  

 

Everyone  contributed  to  the  calculations  required  for  every  section.    

The  following  members  wrote  the  Sections:    

β€’ Luke  Guirguis:  Sections  1,  3  and  4  

β€’ Jose  Sepulveda:  Sections  1,5,  and  6  

β€’ Sean  Godinez:  Sections  1,  2,    and  8  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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8.0 REFERENCES  

β€œAtlas  V  551.”  Spaceflight  101.  n.d.  Web.  13  Mar.  2014.  

http://www.spaceflight101.com/atlas-­‐v-­‐551.html  

Brown,  Charles  D.  Elements  of  Spacecraft  Design.  Web.  13  Mar.  2014.  

Brown,  Charles  D.  Spacecraft  Mission  Design.  2nd  ed.  Web.  13  Mar.  2014.  

Curtis,  Howard  D.  ,  Orbital  Mechanics  for  Engineering  Students  Third  Edition    

Goodman  H.S.  Pre-­‐Flight  Interplanetary  Mission  Analysis.  21  Jan.  1969.  Web.  13  

  Mar.  2014  

β€œHORIZONS  System.”  Solar  System  Dynamics.  Jet  Propulsion  Lab,  n.d.  Web.  13  Mar.  

2014.  http://ssd.jpl.nasa.gov/?horizons  

β€œJuno  Spacecraft  Information.”  Spaceflight  101.  n.d.  Web.  13  Mar.  2014.  

  http://www.spaceflight101.com/juno-­‐spacecraft-­‐information.html  

 

     


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FLUID MECHANICS 1. Division of Fluid Mechanics - … · 1 FLUID MECHANICS 1. Division of Fluid Mechanics Hydrostatics Aerostatics Hydrodynamics Gasdynamics v velocity p pressure ρ

orbital molekul

orbital molekul

Orbital mechanics/Saros/Quiz 1/Testbank Orbital mechanics/Saros/Quiz 1/Testbank Wikiversity 1/26

Orbital mechanics/Saros/Quiz 1/Testbank Orbital mechanics/Saros/Quiz 1/Testbank Wikiversity 1/26

Orbital AtΓ³mico

Orbital AtΓ³mico

Agitador tΓ©rmico Matrix Orbital /Orbital

Agitador tΓ©rmico Matrix Orbital /Orbital

Soldadora Orbital Maus

Soldadora Orbital Maus

2.29 Numerical Fluid Mechanics Fall 2011 - … · 2.29 Numerical Fluid Mechanics Fall 2011 . 2.29 . ... 2.29 Numerical Fluid Mechanics . Project: There will be a final project for

2.29 Numerical Fluid Mechanics Fall 2011 - … · 2.29 Numerical Fluid Mechanics Fall 2011 . 2.29 . ... 2.29 Numerical Fluid Mechanics . Project: There will be a final project for

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