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Glom
2003-Feb-24, 07:08 PM
Been reading Fundamentals of Astrodynamics by R R Bate, D D Muellar and J E White.

I finally managed to work through the section on the prediction problem. I even managed to write a program on my graphics calculator that would do the job. I tested it on the EXAMPLE PROBLEM on page 210 and got more or less the right answer (they said they were using slide rules whereas I was using a modern graphics calculator that can operate up to fifteen decimal places so I don't mind being a few tenths of a percent off from their answer).

I had a problem with Exercise 4.5 on page 223.



A radar ship at 150 W on the equator picks up an object directly overhead. Returns indicate a position and velocity of:

r<sub>0</sub> = 1.2I DU
v<sub>0</sub> = .1I + J DU TU<sup>-1</sup>

Four hours later another ship at 120 W on the equator spots the same object directly overhead. Find the values of f, g, f(dot) and g(dot) that could be used to calculate position and velocity at the second sighting.
(Partial Ans. f(dot) = -.625)


Just for reference, f, g, f(dot) and g(dot) are scalars that satisfy the equns:
r = fr<sub>0</sub> + gv<sub>0</sub>
v = f(dot)r<sub>0</sub> + g(dot)v<sub>0</sub>

In the book, we derived equations for f, g, f(dot) and g(dot) in terms of the universal variable x. x must be obtained by use in the Newton iteration. The problem is that when I entered all the data into my program the iteration refused to converge. I checked it manually and the values just seemed to rise at pretty much a constant rate.

There is also the issue of the coordinates of the observing ships. They seem a bit like a red herring because for this method all you need is r<sub>0</sub>, v<sub>0</sub> and t.

Has anyone, who's read the book, tried and succeeded at this question?

daver
2003-Feb-24, 07:46 PM
On 2003-02-24 14:08, Glom wrote:
Been reading Fundamentals of Astrodynamics by R R Bate, D D Muellar and J E White.
...
There is also the issue of the coordinates of the observing ships. They seem a bit like a red herring because for this method all you need is r<sub>0</sub>, v<sub>0</sub> and t.

Has anyone, who's read the book, tried and succeeded at this question?


I read the book several years ago, some of the sections i skipped over. The problem doesn't look familiar, this is probably one of them.

The coordinates of the ships seem like they should be moderately useful--they tell you that the satellite has moved 90 degrees in four hours. At the very least, it's going to peg it as being in an equatorial orbit, instead of you just getting lucky and observing it when it happened to cross the equator.

Glom
2003-Feb-24, 08:02 PM
The fact that there is no K component to the r or v vectors indicates an equatorial orbit.

As described in chapter 2, r and v at one particular epoch is enough to describe the exact shape of the orbit. Therefore, I don't see how the position of the observation is important.

daver
2003-Feb-24, 09:20 PM
On 2003-02-24 15:02, Glom wrote:
The fact that there is no K component to the r or v vectors indicates an equatorial orbit.

As described in chapter 2, r and v at one particular epoch is enough to describe the exact shape of the orbit. Therefore, I don't see how the position of the observation is important.



Sorry, i should have looked at your post closer. I was thinking that the only info available from the radar was r and dr.

From the examples i remember working through, they are in the habit of giving unnecessary info.

Glom
2003-Feb-24, 09:52 PM
That's why for the moment I have dismissed that information. The problem is that since the Newton iteration won't converge, something else must be up.

Incidentally, as a test of my new program, I tried an extremely simplified example of my own. A circular orbit of radius 1 DU, which means a speed of 1 DU TU<sup>-1</sup>. The period is obviously 2pi TU from the equation for the period. Obviously, half way through that, the orbiter would be on the other side of the orbit and that would be pi TU later.

So, with the input values of:
r<sub>0</sub> = 1I DU
v<sub>0</sub> = 1J DU TU<sup>-1</sup>
t = pi TU

The output values would be expected to be:
r = -1I DU
v = -1J Du TU<sup>-1</sup>

However, entering that into my program, I got the right velocity but the radius had a J component of 10<sup>-14</sup>. I think an accuracy thing is to blame there.

Glom
2003-Feb-25, 04:57 PM
Today, reading more and realising that my program can only deal with closed orbits, I designed a new one. This one had no problem with that iteration but the value of f-dot that came from it was -0.05 and not -0.6.