Orbital elements - what's wrong with my calculation?

[dominicfhk]

Homework Statement

Ok so this satellite is orbiting the earth. The position vector is r = -0.707i + 0.707j + 0k and velocity vector is v = 0i + 0.5j + 0k. To find Vo (the angle between vector n and e), my book said the formula to use is arccos{(e dot r)/(|e||r|)}. However, I got 98 degree by using this formula while the correct answer is 173 degree.

Homework Equations

arccos{(e dot r)/(|e||r|)}

The Attempt at a Solution

I got the e vector as -.5304i-0.707j+0k, and therefore |e|=-.3535
r vector was given as -0.707i+0.707j+0k, and so |r|=0.9998
"e dot r" = -0.125 and "|e||r|" = 0.884
so arccos{(e dot r)/(|e||r|)} = 98 degree, which does not match the correct answer.

This is from the textbook "Fundamentals of Astrodynamics" problem 2.2 on P.114
http://aeroden.files.wordpress.com/2011/12/fundamentals_of_astrodynamics.pdf

Thank you so much.

 

[oli4]

Hi dominicfhk,

Well, first of all, there are several errors in your calculations, but you didn't post the original equations correctly anyway. I don't really know how you could even get some of those results.
In any case, your pdf is wrong too.
By following it, you would get n as a null vector.

I fiddled around and found the correct values for the exercise that give me the same answers as in the book.
Well, almost, I don't get the exact same numerical answers because of the precision involved, this is because I write √2/2 instead of .707, it's much cleaner for working with the equations and I am quite certain that this is where .707 comes from.

So try with those values instead:
R=(DU√2/2)(-I+K)
V=DU/2TU K

this gives
ρ=DU/8 (as expected)
e=5√2/8≈.883 (vs .885 in the book)
v0=arcos(-7√2/10)≈171.87° (vs 173° in the book)

Cheers...

 

[gneill]

Homework Statement

Ok so this satellite is orbiting the earth. The position vector is r = -0.707i + 0.707j + 0k and velocity vector is v = 0i + 0.5j + 0k. To find Vo (the angle between vector n and e), my book said the formula to use is arccos{(e dot r)/(|e||r|)}. However, I got 98 degree by using this formula while the correct answer is 173 degree.

Homework Equations

arccos{(e dot r)/(|e||r|)}

The Attempt at a Solution

I got the e vector as -.5304i-0.707j+0k, and therefore |e|=-.3535

For the eccentricity vector, check the sign of the j component and the calculated magnitude; how can a magnitude be negative

r vector was given as -0.707i+0.707j+0k, and so |r|=0.9998

As pointed out by Oli4, the 0.707's are probably meant to be $\sqrt(2)/2$, else a radius of 0.9998 implies that the satellite is orbiting below the Earth's surface! Thus you can reasonably assume the radius magnitude to be 1.0 .

"e dot r" = -0.125 and "|e||r|" = 0.884

The value for |e||r| looks okay, but check your calculation for e dot r. I'm not sure how you managed to get the value for |e||r| correct when your magnitude of e above is wonky.

so arccos{(e dot r)/(|e||r|)} = 98 degree, which does not match the correct answer.

Once you straighten out the details for the eccentricity vector and its magnitude, all should be well. Don't worry overly much about being bang on in the third decimal place; it's possible that some of the problem results were calculated by slide rule!

This is from the textbook "Fundamentals of Astrodynamics" problem 2.2 on P.114
http://aeroden.files.wordpress.com/2011/12/fundamentals_of_astrodynamics.pdf

Thank you so much.