Transfer
Calculations
Hohman
Transfer
Free
Return Trajectory
Spiral
Summary
of Numerical Data
Launch
Windows
Calculations
Possible
Launch Dates
Graphical
Comparison
Sources
Calculations
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,
= 
= 
m
= 
= 
m
r = 
=
polar ellipse equation
= 
= 
= 
= 
= 
m
= 
---> 
= 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.
= 
= 
=
= 
= 
= 
=72439m/s
=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=7.2km/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
References:
Kelppner, Daniel and Kolenkow, Robert
J. An Introduction To Mechanics. McGraw Hill, Inc., Boston: 1973.
Mars Academy. www.marsacademy.com.
4/10/2000.
