Now that we know how to get to another planet, how do
we know when to head in the right direction. If we don't launch the
spacecraft on the elliptical orbit around the Sun at the right moment,
the spacecraft will miss the planet rendezvous and continue on its orbit
around the Sun.

Luckily, NASA is pretty careful about planning their
space flights as they proved with the Galileo Spacecraft as demonstrated
below. What NASA refers to as a launch window
generally has to do with some considerations such as these:

if the spacecraft is to reach a position in another planet's
orbit at the same time as that planet reaches it, there must be a certain
relationship between the position of the Earth and the position of the
planet at the time of launch. By giving slightly larger or smaller boosts
at each of the steps we have talked about in this course, it is usually
possible to launch a spacecraft any time within an interval of a few
days or a week (a.k.a. a launch window in time) and still manage a rendezvous
without using excessive amounts of fuel. If a launch window is missed,
it may be quite a while for the next launch window to be available.

So, now the question is, when do we leave?

We have already discussed the type of orbit (elliptical)
and the size of the orbit (a perihelion of 1 AU and an aphelion of 5.2
AU). The obvious question that still needs to be answered is, "how
long does it take for the spacecraft to get from Earth to Jupiter?"
With that bit of information, we can compute where Earth and Jupiter
must be in space, relative to each other, to establish when a launch
window is available.

For the answer of how long it will take a spacecraft to
reach Jupiter, we need to go back to Kepler's Third Law of planetary
motion. In mathematical form it is P^{2} = a^{3} where we need to be careful
to use years for P and AU for a.

Remember what the elliptical orbit of the spacecraft looks
like when it heads to Jupiter.

We can find the long axis - the major axis - by adding
5.2 and 1 to get 6.2. The semi-major axis is just half that length,
or 6.2 / 2 = 3.1 AU

The period is therefore:

P^{2} = a^{3}

P^{2} = (3.1)^{3}

With a handy calculator we can solve this: P = 5.5 years
for one total orbit. To reach Jupiter (at aphelion in the transfer orbit)
will take just half that time, or 2.75 years (one way on a 5.5 year
round trip). If you go back and look at the launch date and arrival
date for the Galileo mission to Jupiter you'll find that this is indeed
how long it took to get there.

Finally, to finalize the plan, you have to find a date
when the arrangement of Earth and Jupiter are right for the launch of
the spacecraft with the knowledge of where the planets are relative
to each other. For each possible launch date we can easily figure out
where Jupiter will be 2.75 years later, and where the mission will be
in Jupiter's orbit; the only good launch dates are those where the spacecraft
and Jupiter end up at the same place.

You'll be learning how to use this information in Unit
7, where you will plan your own mission to a planet!