Astronomy Principles and Practice Fourth Edition.pdf

196 Celestial mechanics: the many-body problem
Figure 14.7. Description of two co-planar orbits.
One of the first men to study interplanetary orbits in which instruments could be sent from Earth
to other planets was Dr Walter Hohmann. In the 1920s he showed that a particular type of orbit was
the most economical in rocket fuel expenditure. We can best appreciate his argument by studying the
situation presented in the next section.
14.8.2 Transfer between circular, coplanar orbits about the Sun
In figure 14.7, we have two circular, coplanar orbits of radii a 1 and a 2 astronomical units (AU).
Consider the energy C 1 of a particle in the orbit of radius a 1 . It is given by using equation (13.20).
We obtain
1
2 V 2 1 − µ a 1
= C 1
where V 1 is the velocity in the orbit. By equation (13.30),
V 2 1 = µ a 1
.
Then,
C 1 =− µ .
2a 1
Similarly, the energy C 2 in the orbit of radius a 2 is given by
C 2 =− µ .
2a 2
Now a 2 > a 1 ,sothatC 1 < C 2 , in other words a change of orbit would be a change of energy.
This change of energy is brought about by the rocket engine. By imparting a velocity increment V to
the rocket, it changes its kinetic energy and, hence, its total energy. Hohmann studied how this could
be most effectively done.
The kinetic energy in equation (13.20) is the term 1 2 V 2 . In figure 14.8, we see the original velocity
V . The increment velocity V could be applied as shown in figure 14.8 in such a way that only the
direction of the vehicle’s velocity is changed (from V to V ′ ) but not the magnitude (V ′ = V ). Inthat
case, the new kinetic energy 1 2 V ′2 would be the same as the pre-burn kinetic energy. No change in total
energy would be achieved.