M
mcconnel
Guest
I spent much of my long run Saturday afternoon trying to make sense of the Columbia tragedy. Like
just about everyone else I failed in that, but the effort did make me consider some interesting
issues about motion -- getting from here to there. Inasmuch as this group is very much interested
in getting from here to there, hopefully sooner rather than later, a discussion of these seems
somewhat on topic.
One of the first things I wondered when I heard about the probable damage from a piece of foam
striking the left wing during launch was "why didn't they look and see if the damage were serious
enough to warrant corrective action?" I had in mind some scheme involving hooking up with the space
station and having all 7 astronauts camp out there while waiting to be rescued.
As shuttle program director Ron Dittemore explained, this idea is fraught with difficulties and
illustrates misconceptions many people have about motion in orbit. At any given moment the shuttle
and space station were many thousands of miles apart, but what are thousands of miles to a ship
capable of moving at thousands of miles per hour? Unfortunately, catching up with the distant space
station isn't quite as simple as catching another car on the freeway. For one thing, the station is
in a different orbit altogether; the analogy of catching a car on a different freeway is somewhat
closer to the truth, but even that is misleading.
Suppose, e.g., you first match your orbit with the space station's (sort of like getting on the
correct freeway,) and then attempt to "catch up" with it by speeding up in the direction along the
orbit. If you don't speed up, you'll never catch up, but if you do try to speed up, you'll merely
move into a higher orbit! When you finally pull abreast of the space station you may find yourself
many miles above it.
To put it another way, a race in orbit would have a curious feature: try to speed up and you'd end
up going uphill!
To take an even more extreme example, suppose astronomers discovered a comet streaking toward the
sun from the distant edges of the solar system on a hyperbolic orbit. Hyperbolic means that the
comet is moving too fast to ever fall into permanent orbit around the sun. (This is not uncommon for
comets. Many of them visit the inner solar system only once, whipping around the sun and thereafter
returning to interstellar space, never to be seen again.) Let's say someone proposes sending a
spacecraft out to meet the comet in order to study it, take pictures, or even land and have a look
around. Simple, you say. Just send the spacecraft out in such a way that it will arrive at some
point at the same time the comet does. Unfortunately, if you do this is a naive way, the comet will
whip by you so fast you will probably not even see it. What you really have to do is to match your
orbit with the comet's. This involves at least accelerating to the enormous velocity needed to enter
a hyperbolic (escape) orbit around the sun.
The key, then, to understanding rendezvous manouevers in space is the concept of matching orbits.
But, as explained above, it is not enough just to match orbits. You must do so in exactly the right
way, so that at the instant you reach the correct orbit you also find your goal object in exactly
the same spot on that orbit as you are. Needless to say, this takes some very careful planning (i.e,
mathematics,) to arrange.
--
************************************************************************
Terry R. McConnell Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150 [email protected]
http://barnyard.syr.edu/~tmc Are we there yet?
************************************************************************
just about everyone else I failed in that, but the effort did make me consider some interesting
issues about motion -- getting from here to there. Inasmuch as this group is very much interested
in getting from here to there, hopefully sooner rather than later, a discussion of these seems
somewhat on topic.
One of the first things I wondered when I heard about the probable damage from a piece of foam
striking the left wing during launch was "why didn't they look and see if the damage were serious
enough to warrant corrective action?" I had in mind some scheme involving hooking up with the space
station and having all 7 astronauts camp out there while waiting to be rescued.
As shuttle program director Ron Dittemore explained, this idea is fraught with difficulties and
illustrates misconceptions many people have about motion in orbit. At any given moment the shuttle
and space station were many thousands of miles apart, but what are thousands of miles to a ship
capable of moving at thousands of miles per hour? Unfortunately, catching up with the distant space
station isn't quite as simple as catching another car on the freeway. For one thing, the station is
in a different orbit altogether; the analogy of catching a car on a different freeway is somewhat
closer to the truth, but even that is misleading.
Suppose, e.g., you first match your orbit with the space station's (sort of like getting on the
correct freeway,) and then attempt to "catch up" with it by speeding up in the direction along the
orbit. If you don't speed up, you'll never catch up, but if you do try to speed up, you'll merely
move into a higher orbit! When you finally pull abreast of the space station you may find yourself
many miles above it.
To put it another way, a race in orbit would have a curious feature: try to speed up and you'd end
up going uphill!
To take an even more extreme example, suppose astronomers discovered a comet streaking toward the
sun from the distant edges of the solar system on a hyperbolic orbit. Hyperbolic means that the
comet is moving too fast to ever fall into permanent orbit around the sun. (This is not uncommon for
comets. Many of them visit the inner solar system only once, whipping around the sun and thereafter
returning to interstellar space, never to be seen again.) Let's say someone proposes sending a
spacecraft out to meet the comet in order to study it, take pictures, or even land and have a look
around. Simple, you say. Just send the spacecraft out in such a way that it will arrive at some
point at the same time the comet does. Unfortunately, if you do this is a naive way, the comet will
whip by you so fast you will probably not even see it. What you really have to do is to match your
orbit with the comet's. This involves at least accelerating to the enormous velocity needed to enter
a hyperbolic (escape) orbit around the sun.
The key, then, to understanding rendezvous manouevers in space is the concept of matching orbits.
But, as explained above, it is not enough just to match orbits. You must do so in exactly the right
way, so that at the instant you reach the correct orbit you also find your goal object in exactly
the same spot on that orbit as you are. Needless to say, this takes some very careful planning (i.e,
mathematics,) to arrange.
--
************************************************************************
Terry R. McConnell Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150 [email protected]
http://barnyard.syr.edu/~tmc Are we there yet?
************************************************************************