# Computing the local orbital frame vector directions in a geocentric reference frame

In the context of the study of a cubesat (little satellite), I am asked to define the local orbital frame and compute the local orbital frame vector directions as a function of time :

Here the context : Anyone could give me some tracks or links to find the answer of part 2.1 Local orbital frame ? this would be kind.

Regards

• The radial or local vertical vector $$R$$ is just the position vector from the center of the Earth to the spacecraft.
• The in-track, transverse, or local horizontal vector $$I$$ is the vector in the orbital plane that is both prograde and perinduclar to the radial vector. It is not, in general, the velocity vector. (The in-track and velocity vectors are coincident if and only if the flight path angle is zero, i.e., you have one of a) a circular orbit, b) the spacecraft is at apoapsis, or c) the spacecraft is at periapsis). You can construct it by applying the Graham-Schmidt Process to the position and velocity vectors; $$I = v - \frac{R\cdot v}{R \cdot R}R$$.
• The cross-track or normal vector $$C = R \times I$$ completes the right-handed set.