Re: Continuous thrust changes in orbits
- From: Nyrath the nearly wise <nyrath@xxxxxxxxxxxxxxxxxxxxxx>
- Date: Wed, 04 Apr 2007 23:54:14 -0400
Raghar wrote:
It looks like a continuous thrust could make computation of change in
orbits quite difficult.
Do you have any idea about how could be these calculated sensibly?
There was a technique mentioned in Pournelle and Sheffield's novel
HIGHER EDUCATION. In the novel, the apprentices are on a continuous
thrust spacecraft heading to the asteroid belt to an asteroid called
CM-26. They are given a problem of estimating the transit time.
One of the apprentices recalls that the teacher mentions that
the acceleration due to the Sun at Earth's orbit is about one
one-thousandth of a gee, and at the asteroid belt it is closer
to one ten-thousandth of a gee. Since the ship is accelerating
at about one quarter gee, he realizes that the sun's
acceleration can be ignored.
This means one can assume that the ship is traveling in
a straight line, and not in some kind of loopy orbit.
The main unknown is the distance to the destination.
From the novel:
"Alice, we don't need to know a whole slew of orbital theory. Our own ship's acceleration is so high, we can get a good value for our travel time by assuming that we travel in a straight line and ignoring everything else. All we need is our own acceleration, and the distance we have to go."
"But we don't know the distance -- CM-26 keeps moving."
"Sure it does." The boredom and frustration had been swept away by excitement and certainty. Rick suddenly had a clear mental picture in his head. "See, here's what we do. We make two tables. The first has two columns. It shows times, say, every hour from the time we started, in the left column. The right column shows the distance from CM-26, where we started, to CM-26 for the corresponding time. It has to be a table, because CM-26 keeps moving. The information to make that distance table comes right out of the coordinates given to us in the problem packet."
"But we don't know the travel time -- that's what we're supposed to find out!"
"I know. Let me finish. Now, we make another table. This one also has two columns. The left column is the same list of times, in hours. The right column says how far the ship goes in that much time, assuming that we accelerate for half the time, and decelerate for the other half. We already had the formula in class for the distance traveled in a given time with a given constant acceleration. It's just half the time multiplied by the final speed."
"But we don't know the final speed!"
"Yes we do -- at least, we know the formula for it. We had it on a test, it's just the acceleration multiplied by the time."
"If you say so. But I still don't know what we do next."
"We're almost finished. We have two time/distance tables, right? Now we plot them both as curves on one graph. The first curve is the distance to CM-26, hour by hour. The second curve shows the ship's distance traveled, also hour-by-hour. Those two curves cross somewhere -- they have to, or we'd never be able to reach CM26. And the time where they cross is the travel time -- our answer. It's the time when the ship will have gone just far enough to reach the position of CM-26. An approximation, but I bet it will be closer than 10 percent."
.
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