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Scientific Background :

Based on a Runge Kutta simulation with variable mesh-width (using the Hamilton Equations
in a polar coordinate system that sets the centre of total mass [earth+moon] S as centre),
the flight trajectory of the transported object which passes very close to the moon is obtained for the following initial values :

[ v : initial orbital velocity around earth ; r : distance of the spacecraft to S ;
used moon/earth mass ratio of m2/m1 = 0.02
(this allowed the used model with an unguided spacecraft to "hit" the moon better ;
more realistic would be a value of 0.012 = 1/81 ;
one might want to install the automatic circle recognition
inside the object to correct its path during flight ) ;
assumed distance earth <-> moon : 384 000 km (mean distance) ;
used moon orbital period : 27 d = 648 hours ]

possible landing concepts

a)
In order to crash-land an unmanned object on the moon one must reduce its velocity
by about 3500 km/h to enter an orbit whose nearest point to the moon is on the lunar surface.
The object flies over the lunar surface with a relative velocity of about 9500 km/h (hitting a crater incidently or being slowed down by reverse thrust).
b) Realizing a transport to the moon with a soft lunar landing requires in the used simplified model
(masses of spacecraft and other planets were ignored for the gravitational potential ;
velocity changes are assumed to need no time to be executed) the following
velocity changes:
v1 = 39000 km/h
v2 = 3500 km/h
v3 = 9500 km/h
necessary for leaving earth orbit and setting course for the moon
slowing down for lunar orbit
reaching v=0 km/h for a soft - and less destructive - landing on the moon
( all initial data and results for vi are based on computations described in
"Theoretical Physics on the Personal Computer" by E.W. Schmid / G. Spitz / W. Lösch )

The energy- (and with its help the fuel-) consumption for each step vi can be computed with
Ei = 1/2 * mi * va 2 - 1/2 * mi * (va - vi)2 = 1/2 * mi * [ va2 - (va - vi)2 ]
= 1/2 * mi * ( va2 - va2 + 2*va*vi - vi2 ) =
1/2 * mi * ( 2*va*vi - vi2 )

( with :
va : velocity at begin of acceleration step i
mi : base mass of object + mass of actually carried fuel at time of step i
note: the mass change during the acceleration step is ignored in this simplified model )

(this page has been updated on April-01-2001)