Transfer Calculations
  Hohman Transfer
  Free Return Trajectory
  Summary of Numerical Data

Launch Windows
  Possible Launch Dates
  Graphical Comparison


Free Return Trajectory


Let  = period of free return orbit  = 730 days (twice 365)  because when the spacecraft returns to its starting point, Earth has orbited the sun twice and is in the same spot.
~ therefore,
r = = polar ellipse equation

= 215 days
= G (conservation of energy)
= = 110418 m/s
= v of free-return at Mars = =77912 m/s
= 110418-94383 =16305 m/s
For Mars, this equation involves vectors as the spacecraft is not moving tangent to Mars' orbit when it arrives.


=28685 m/s
== 28966 m/s
see work for Hohmann transfer:
D=???????m/s - 7842 m/s =11.6 km/s
D= ?=29038m/s-1449m/s = 27.6 km/s

Launch Windows, Durations of Flights, and Stay on Mars

The following are calculations detailing the times of transfer for both the Hohmann and Free Return Trajectories.

Additional Definitions of Variables Let

= period of Earth's orbit = 365.25 days = period of Mars' orbit

m = Earth's semi-major axis

m = Mars' semi-major axis

Q = = Earth's lead on 3/27/1997

Hohmann Transfer Launch Windows, Duration of Flight, and Stay on Mars Calculations

days= 258.6 days = =timeof Hohmann transfer

At beginning of transfer, Mars must lead Earth by

= 258.6 days = 

At the end of the transfer, Mars leads Earth by

Mars to Earth:

days = 258.6 days = 


At the beginning of the transfer, Mars must lead by

= degrees Earth moves relative to Mars during the stay = 

=period of Earth relative to Mars

f = frequency

= 780.1 days

=455 days

Free Return Trajectory Launch Windows, Duration of Flight and Stay on Mars

days =215 days = 


At the beginning of the transfer, Mars must lead Earth by

days = 215 days 


At the end of the transfer, Mars leads Earth by


Earth to Mars (Hohmann transfer)

At the beginning of the transfer, Mars leads Earth by

=388.1 days

Launch opportunities occur every  for both transfers, and they can be determined by similar methods to those shown above if is known for some .

The following chart shows the results of these calculations for Mars ahead of earth on 3.27.1997.

The comments are based on the solar flare and dust storm chart that follows.

Launch opportunities for free return occur 455+258-388-215 = 110 days later than those for the Hohmann transfers, and landings occur 455-388 =67 days later than those of the Hohmann transfer.

Summary of Above Calculations

Hohmann Transfer (both directions)

Earth to Mars:

D=6.6 km/s 

D=7.2 km/s

Mars to Earth:

D=6.6 km/s 


D= 27.6 km/s

= 258 days

=455 days

= 258 days

Free Return (to Mars, Hohmann return)

Earth to Mars:

D=11.6 km/s

D=27.6 km/s

Mars to Earth:

D=7.2 km/s

D=6.6 km/s

=53 km/s

= 215 days

=386 days

=258 days

Kelppner, Daniel and Kolenkow, Robert J. An Introduction To Mechanics. McGraw Hill, Inc., Boston: 1973.
Mars Academy. 4/10/2000.

mitCopyright © 2000 Massachusetts Institute of Technology
Comments and questions to Last updated: 10 December, 2000