Editing Spacecraft flight dynamics (section)

=====Altitude change by bi-elliptic transfer=====
[[File:Bi-elliptic transfer.svg|thumb|A bi-elliptic transfer from a low circular starting orbit (dark blue) to a higher circular orbit (red)]]
A slightly more complicated altitude change maneuver is the [[bi-elliptic transfer]], which consists of two half-elliptic orbits; the first, posigrade burn sends the spacecraft into an arbitrarily high apoapsis chosen at some point <math>r_b</math> away from the central body. At this point a second burn modifies the periapsis to match the radius of the final desired orbit, where a third, retrograde burn is performed to inject the spacecraft into the desired orbit.<ref name="Curtis">{{Cite book | last = Curtis | first = Howard | title = Orbital Mechanics for Engineering Students | page = 264 | publisher = [[Elsevier]] | year = 2005 | isbn = 0-7506-6169-0 | url = https://books.google.com/books?id=6aO9aGNBAgIC}}</ref> While this takes a longer transfer time, a bi-elliptic transfer can require less total propellant than the Hohmann transfer when the ratio of initial and target orbit radii is 12 or greater.<ref>{{cite journal
| last1       = Gobetz
| first1      = F. W.
| last2      = Doll
| first2     = J. R.
| date       = May 1969
| title      = A Survey of Impulsive Trajectories
| journal    = AIAA Journal
| publisher  = [[American Institute of Aeronautics and Astronautics]]
| volume     = 7
| issue      = 5
| pages      = 801–834
| doi        = 10.2514/3.5231| bibcode= 1969AIAAJ...7..801D
}}</ref><ref>{{Cite book
 | first = Pedro R.
 | last = Escobal
 | title = Methods of Astrodynamics
 | location = New York
 | publisher = [[John Wiley & Sons]]
 | year = 1968
 | isbn = 978-0-471-24528-5
}}</ref>

Burn 1 (posigrade):
<math display="block">\Delta v_1\ = {v_p}_1 - v_1</math>
Burn 2 (posigrade or retrograde), to match periapsis to the target orbit's altitude:
<math display="block">\Delta v_2\ = {v_a}_2 - {v_a}_1</math>
Burn 3 (retrograde):
<math display="block">\Delta v_3\ = v_2 - {v_p}_2</math>