The paper calculates the delta-V requirements to launch a spacecraft into Low Earth Orbit and execute Hohmann transfers to the International Space Station orbit and Geosynchronous Orbit. These were found to be 8.03 kms-1, 0.12 kms-1 and 3.94 kms-1, respectively. The paper also includes researched values for the delta-V requirements to travel to the Moon and Phobos. The paper aims to highlight the fact that delta-V requirements for manoeuvres beyond Low Earth Orbit are considerably lower than achieving Low Earth Orbit.Â

Back, Ashley

Brown, Gregory

Hall, Ben

Turner, Stuart

English

University of Leicester Open Journals

Journal	  of	  Physics	  Special	  Topics	   1  P1_10	  Delta-­‐V	  Requirements	  of	  Spaceflight	  	  	  Back	  A,	  Brown	  G,	  Hall	  B	  and	  Turner	  S	  Department	  of	  Physics	  and	  Astronomy,	  University	  of	  Leicester,	  Leicester,	  LE1	  7RH.	  November	  23rd,	  2011	  Abstract	  The	  paper	  calculates	  the	  delta-­‐V	  requirements	  to	   launch	  a	  spacecraft	   into	  Low	  Earth	  Orbit	  and	  execute	  Hohmann	  transfers	  to	  the	  International	  Space	  Station	  orbit	  and	  Geosynchronous	  Orbit.	  These	   were	   found	   to	   be	   8.03	   kms-­‐1,	   0.12	   kms-­‐1	   and	   3.94	   kms-­‐1,	   respectively.	   The	   paper	   also	  includes	  researched	  values	  for	  the	  delta-­‐V	  requirements	  to	  travel	  to	  the	  Moon	  and	  Phobos.	  The	  paper	   aims	   to	   highlight	   the	   fact	   that	   delta-­‐V	   requirements	   for	  manoeuvres	   beyond	   Low	   Earth	  Orbit	  are	  considerably	  lower	  than	  achieving	  Low	  Earth	  Orbit.	  	  	  	  Introduction	  The	  past	  ten	  years	  have	  seen	  the	  beginnings	  of	  space	  tourism	  with	  the	  first	  space	  tourist,	  Dennis	  Tito,	  spending	  eight	  days	  in	  space	  onboard	  the	  International	  Space	  Station	  at	  the	  cost	  of	  £14m	  [1].	  Since	  this	  pioneering	  flight	  there	  have	  been	  six	  other	  space	  tourists,	  each	  paying	  between	  $20m	  and	  $30m	  for	   the	  privilege	  to	  stay	  onboard	  the	   International	  Space	  Station	  [2].	   In	  all	  cases	  the	  tourists	  have	   travelled	   to	   the	   International	   Space	   Station	   onboard	   the	   Russian	   Soyuz	   [2],	   arranged	   by	   the	  company	  Space	  Adventures	  [3].	  The	  key	  to	  launching	  a	  spacecraft	  and	  manoeuvring	  a	  spacecraft	  in	  the	  zero-­‐g	  environment	  is	  to	  understand	   the	  delta-­‐V	   requirements	   per	   action.	  As	   these	  manoeuvres	   are	   commonly	   dictated	  by	  the	   amount	   of	   fuel	   the	   spacecraft	   has,	   it	   follows	   that	   the	   cost	   of	   a	   mission	   into	   space	   depends	  somewhat	  upon	  the	  delta-­‐v	  requirements	  of	  said	  mission.	  With	  the	  recent	  surge	  of	  interest	  in	  space	  tourism	  by	  various	  companies,	   this	  paper	   looks	  at	   the	  delta-­‐V	  requirements	  to	   launch	  a	  spacecraft	  into	  Low	  Earth	  Orbit	  and	   to	  conduct	  Hohmann	   transfers	   to	   the	   International	   Space	  Station	  and	   to	  Geosynchronous	  Orbit.	  The	  values	  for	  the	  required	  delta-­‐V’s	  for	  transfers	  to	  the	  Moon	  and	  Phobos	  are	  researched.	  	  Theory	  To	   calculate	   the	   delta-­‐V	   required	   for	   a	   Hohmann	   transfer,	   the	   Vis-­‐Viva	   Integral	   is	   used	   to	  determine	   the	   velocities	   of	   the	   initial	   circular	   orbit,	   elliptical	   Hohmann	   transfer	   orbit	   and	   final	  circular	  orbit.	   ∆𝑣! = 𝑣!"# − 𝑣! = !!!! − !!!!!!! − !!!	  	  	  	  	  	  	  (1)[4]	  ∆𝑣! = 𝑣!! − 𝑣!"# = !!! − !!!! − !!!!!!!	  	  	  	  	  	  	  (2)[4]	  Where	  ∆𝑣!	   and	  ∆𝑣!	   are	   the	   required	  delta-­‐V’s	   to	  manoeuvre	   the	  orbit	   of	   the	   spacecraft	   into	   the	  elliptical	   Hohmann	   transfer	   orbit	   and	   the	   final	   circular	   orbit,	   respectively.	   𝑣! 	   and	   𝑣!! 	   are	   the	  velocities	   of	   the	   initial	   and	   final	   circular	   orbits,	   respectively.	   𝑣!"# 	   and	   	   𝑣!"#	   are	   the	   velocities	   at	  periapsis	  and	  apoapsis	  of	  the	  elliptical	  Hohmann	  transfer	  orbit,	  respectively.	  	  𝑟!	  and	  𝑟!	  are	  the	  radii	  of	  the	  initial	  and	  final	  circular	  orbits,	  respectively.	  𝜇	  is	  the	  gravitational	  parameter	  of	  the	  body	  being	  orbited	  (3.98	  x	  10-­‐8	  m3s-­‐2)	  [5].	  The	  total	  delta-­‐V	  required	  for	  the	  Hohmann	  transfer	  is	  given	  by	  ∆𝑣 = ∆𝑣! + ∆𝑣!.	  	  	  	  	  	  	  (3)	  Analysis	  Using	  equations	  (1),	  (2)	  and	  (3)	  the	  delta-­‐V	  requirements	  were	  calculated	  to	  travel	  from	  the	  radius	  of	  the	  Earth	  (6370	  km):	  to	  Low	  Earth	  Orbit	  (200	  km),	  the	  International	  Space	  Station	  (400	  km	  [6])	  and	  Geosynchronous	  Orbit	  (36000	  km).	  These	  values	  are	  shown	  in	  table	  1	  below.	  P1_10	  Delta-­‐V	  Requirements	  of	  Spaceflight,	  November	  23rd,	  2011.	  2  	  Manoeuvre	   𝒓𝟏	  (km)	   𝒓𝟐	  (km)	   ∆𝒗	  (kms-­‐1)	  Launch	  to	  LEO	   6370	   6570	   8.03	  LEO	  to	  ISS	   6570	   6770	   0.12	  LEO	  to	  GEO	   6570	   42000	   3.94	  Table	  1:	  Table	  listing	  values	  for	  𝑟!,	  𝑟!	  and	  the	  resultant	  ∆𝑣	  for	  the	  Hohmann	  transfers	  from	  launch	  to	  Low	  Earth	  Orbit	  (LEO),	  LEO	  to	  the	  International	  Space	  Station	  (ISS)	  and	  LEO	  to	  Geosynchronous	  Orbit	  (GEO).	  	  	  Table	  1	  lists	  the	  delta-­‐V	  requirements	  for	  the	  Hohmann	  transfer	  orbits.	  It	  was	  assumed	  that	  the	  initial	  velocity	  of	  the	  spacecraft	  on	  Earth	  was	  0	  kms-­‐1	  and	  an	  ideal	  case	  was	  assumed	  for	  launch,	  so	  atmospheric	   drag	   was	   neglected	   and	   only	   a	   horizontal	   velocity	   component	   considered.	   However,	  when	  atmospheric	  drag	  is	  taken	  into	  account	  and	  a	  vertical	  launch	  assumed,	  the	  velocity	  given	  to	  the	  spacecraft	  due	  to	  the	  Earth’s	  rotation	  offsets	  the	  velocity	  required	  to	  overcome	  these	  issues.	  	  Values	  for	  orbital	  transfers	  to	  the	  Moon	  and	  Phobos	  were	  researched.	  It	  was	  found	  that	  a	  delta-­‐V	  of	   15.2	   kms-­‐1	   is	   required	   to	   travel	   from	   the	   surface	  of	   Earth	   to	   the	   lunar	   surface	   [7].	  Whereas,	   to	  travel	  from	  the	  surface	  of	  Earth	  to	  the	  surface	  of	  Phobos;	  a	  delta-­‐V	  of	  14.0	  kms-­‐1	  is	  required	  [7].	  The	  oddity	  that	  less	  delta-­‐V	  is	  required	  to	  travel	  to	  Phobos	  than	  the	  Moon	  can	  be	  easily	  explained	  by	  the	  fact	  that	  the	  gravitational	  sphere	  of	  influence	  of	  Phobos	  is	  considerably	  smaller	  than	  the	  Moon’s.	  	  Discussion	  There	  are	  other	   factors	  which	  need	  considering	  when	  planning	  missions	   into	   space,	  other	   than	  the	  delta-­‐V	  requirements.	  Manned	  missions	  of	  a	  longer	  duration	  will	  require	  more	  supplies	  such	  as	  food,	  water	  and	  oxygen.	  These	  requirements	  will	  obviously	  increase	  the	  mass	  of	  the	  spacecraft	  and	  hence	  the	  cost	  of	  the	  launch.	  Missions,	  which	   involve	  extended	  travel	  beyond	  Low	  Earth	  Orbit,	  will	  be	  at	  risk	  from	  prolonged	  exposure	  to	  solar	  radiation.	  A	   form	  of	  radiation	  shield	  may	  have	  to	  be	   incorporated,	   this	   to	  would	  add	  to	  the	  cost	  of	  the	  mission.	  Conclusion	  	   Comparing	  the	  results	  in	  table	  1	  and	  the	  values	  obtained	  for	  a	  transfer	  to	  the	  Moon	  and	  Phobos,	  it	  can	  be	  deduced	  that	  the	  most	  costly	  aspect	  of	  space	  travel,	  in	  terms	  of	  delta-­‐V,	  is	  achieving	  a	  Low	  Earth	   Orbit	   about	   the	   Earth.	   Once	   there	   it	   is	   relatively	   inexpensive	   to	   travel	   to	   higher	   orbit	   and	  beyond.	   Therefore,	   the	   space	   tourism	   industry	   could	   look	   at	   the	   possibility	   of	   offering	   cheaper	  “flights”	  from	  Low	  Earth	  Orbit	  to	  more	  distant	  destinations	  compared	  to	  more	  costly	  “flights”	  from	  Earth.	   Alternatively,	   if	   a	   technology	   were	   developed	   which	   would	   enable	   the	   mass	   transit	   of	  customers	   from	  the	  Earth	  to	  Low	  Earth	  Orbit,	   space	  travel	  would	  become	  a	   little	  more	  affordable.	  The	  two	  values	  researched	  for	  the	  delta-­‐V	  requirements	  to	  travel	  to	  the	  Moon	  and	  Phobos	  help	  to	  illustrate	  that	  it	  is	  not	  always	  obvious	  which	  destinations	  require	  the	  least	  delta-­‐V.	  References	  [1]http://news.bbc.co.uk/onthisday/hi/dates/stories/april/28/newsid_2501000/2501015.stm,	  	  accessed	  23/11/2011	  [2]	  http://didyouknow.org/lists/spacetourists/,	  accessed	  23/11/2011	  [3]	  http://www.spaceadventures.com/index.cfm?fuseaction=orbital.Clients,	  accessed	  23/11/2011	  [4]	  University	  of	   Leicester,	  Department	  of	  Physics	  and	  Astronomy,	  PA3613:	   Space	  Systems	  Design,	  Dr.	  Nigel	  Bannister.	  [5]	  http://www.calculator.org/property.aspx?name=gravitational%20constant,	  accessed	  6/12/2011	  [6]	  http://suzymchale.com/ruspace/issorbit.html,	  accessed	  23/11/2011	  [7] http://web.archive.org/web/20070701211813/http://www.pma.caltech.edu/~chirata/deltav.ht	  ml,	  accessed	  23/11/2011	  

P1_10 Delta-V Requirements of Spaceflight

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P1_10 Delta-V Requirements of Spaceflight

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