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I am working on a personal project, and I want to write a program to obtain the RAAN given the angle of the eccentricity vector to the X-axis (ECI) along the XY plane, the declination (angle from XY plane) of the orbiter at apogee, and the inclination of the orbit. I believe that is all I would need to obtain the RAAN, but my efforts have thusfar failed.

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This is not enough information.

All you have is the eccentricity vector and the orbital inclination. (Declination of apoapsis is derivable from the eccentricity vector's direction, as the eccentricity vector points along the line of apsides from apoapsis to periapsis).

For any orbital inclination and chosen eccentricity vector direction (magnitude is irrelevant for these purposes) where the angle between the eccentricity vector and the reference plane is less than the orbital inclination, there are two orbital planes, symmetrical across the plane that contains the eccentricity vector and the reference plane perpendicular, that satisfy the condition.

Two orbital planes with the same eccentricity vector and inclination

This image shows two orbital planes (red and green circles) with the same eccentricity vector direction (gray arrow), and the same orbital inclination (in this case 31.2°), but with Right Ascension of the Ascending Node values 90° apart.

As a result, you'll wind up with two differing values of Right Ascension of the Ascending Node, and will need more information to determine the correct one.

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