Circular prograde orbit homework

[ExoP]

Homework Statement

Hi!

I have a problem with an assignment and need som help. The question is:
You’ve just completed an analysis of where the Space Shuttle must be when it performs a
critical maneuver. You know the shuttle is in a circular prograde orbit and has a position
vector of:
ro= 6275.396Î + 2007.268J + 1089.857K. In 55 minutes, you predict the orbital parameters are :
a = 1.0470357ER ,e = 0.000096, i = 28.5 degrees , M = 278.94688 degrees
Comments: The initial orbit is circular, but the final orbit has eccentricity different from 0, but
it is small, perhaps caused by disturbances.
Is your analysis correct?
Answer with either of the two beginnings:
a) The analysis can’t be correct, because ….
b) The analysis is correct, provided that periapsis has been created at …..

How should I tackle this problem? I really don't have an idea where to start. Should I use the values given and then decide if it is a circular prograde orbit or not? That is my thought. Thanks!

 

[mfb]

Assuming the shuttle is in space and does not accelerate, the orbital parameters do not change. Does the path described by them go through your initial point?

There is one parameter missing, but as the eccentricity is so small I guess we can neglect this.[/size]

 

[ExoP]

I should assume that the initial orbit also has an inclination of 28,5 degree. I probably need to calculate the velocity vector, but how...

 

[gneill]

Hmm. The inclination alone doesn't pin down the orientation of the orbital plane. Looks to me like there are two possible orientations for the normal to the plane that would allow the given position vector to lie in the plane and have a prograde orbit (angular momentum vector lies above the XY plane).

I suppose one might try to construct those two possible normals from the inclination and given position vector. If you posit a unit vector n in the direction of the normal, then its Z-component must be cos(i). If you also take a unit vector in the direction of the position vector (call it r), then $r \cdot n$ = 0 and |n| = 1. Find the n's.

If you've got those normals you can create specific angular momentum vectors from them. All you need is the magnitude of h. Fortunately you've got a circular obit and its radius, so finding the magnitude of the velocity is not a problem...

Speaking of velocity, for a circular orbit the direction of the velocity vector will be at right angles to both the position vector and the orbit plane normal... a trivial vector operation will give you a vector in the right direction.

The orbit parameters for the later given position will let you find the true anomaly for that position. Again, circular orbit, so use the characteristics particular to circular orbits to find the true anomaly at epoch, and to locate the designated "periapsis" (true anomaly is zero there). I guess you could find a way to express the periapsis position in the XYZ frame. Maybe the correct normal places it in a conspicuous place (like towards the ascending node of the orbit, or when it the ship crosses the x-axis). Other than that I can't think of a way to distinguish the two choices or to tell if the given analysis is in error somehow.