Hohmann Transfer Orbit &
Total Travel Time
In the previous unit, we discussed how to get a spacecraft
out of Earth's orbit. But in which direction should we head in order
to meet a planet in all that vast space?
The answer lies in understanding two things: 1 - where
the destination planet is in space and 2 - Kepler's Third Law.
As far as knowing where a destination planet is in space,
we can accomplish that pretty easily by simply keeping track of its
position in the sky and then using what we know about the kind of orbit
it is in to predict where it will be. This kind of prediction has been
pretty accurate for more than a hundred years.
Below is the orbit of the Earth around the Sun (approximated
by a circle, e=0, as we noted at the end of the last subunit).
units, what is the distance of the Earth to the Sun? Click on
the answer button below when you think you know the answer.
In the diagram below we have added the orbit of Jupiter
around the Sun .
To get a spacecraft from the Earth's orbit to Jupiter's
orbit, we need to send the spacecraft on its own orbit around the Sun
that just so happens to correspond to Earth's orbit and Jupiter's orbit.
We do this by creating a spacecraft orbit with a perihelion of 1 AU
(matching the Earth's orbit) and an aphelion of 5.2 AU (corresponding
to Jupiter's orbit).
The spacecraft leaves Earth's orbit and coasts (i.e. does
not use any fuel) out to the orbit of Jupiter. The spacecraft needs
fuel at both ends of the trip to change from the orbit around the Earth
to the elliptical orbit around the Sun, and from the elliptical orbit
to the orbit around Jupiter. This orbit is the orbit that uses the least
possible fuel. It is called the Hohmann transfer orbit.
How do we know this is the cheapest orbit? To get into
this orbit, we fire our rockets while we are at 1AU, near Earth, and
we need a certain speed to get into an orbit that reaches all the way
to Jupiter. If we use less fuel, we never arrive. If we fire our rockets
longer, then the orbit may go beyond our destination. That would also
get us there faster, it turns out, but it would also need more fuel
- not just to get into the orbit, but also at the other end, to match
speed with Jupiter. In the Hohmann orbit we get to Jupiter moving
the same direction but just a bit more slowly than it is (so if we
don't fire our rockets we'll "fall" back to Earth). In a
faster orbit we will arrive on a path that crosses Jupiter's path,
have to use a lot more fuel to match our velocity with Jupiter's.