What is $\phi '$ in orbital mechanics? For the last week or so, I have been teaching myself orbital mechanics within the context of Braeunig's Rocket and Space Technology.

I noticed a symbol, $\phi '$, and was wondering what context that was used in? I think it is an angle measurement, but I know that it is almost equal to the semi-major axis on earth.

From the box labeled Geodetic Latitude, Geocentric Latitude, and Declination, found about half way down after selecting Orbital Mechanics here.
This appears to be the geocentric latitude ($\psi$), which is the angle between the equatorial plane and the point on the surface of the ellipse. It can be calculated from the geodetic latitude (also known as the geographic latitude) ($\phi$) and the eccentricity of the ellipse ($e$):
$$\psi(\phi)=\tan^{-1}\left(\left(1-e^2\right)\tan\phi\right)$$
If $e=0$, then the two lattitudes are the same, because the focus is at the center of the ellipse.



Image courtesy of Wikipedia user Wmc824 under the Creative Commons Attribution 3.0 Unported license.

It might make more sense to use this latitude to describe the position of an orbiting object, but the vector from the focus to the object's position $(r,\theta)$ is perpendicular to the tangential component of the object's velocity vector, meaning that the geodetic latitude may be better.