Believe it or not. There are two Doppler Effects that depend on gamma. The "relativisic" Doppler Effect that occurs at all angles, which is caused by a slowing down of emission processes by motion in the gravitational field of the earth, which has nothing to do with some kind of time dilatation and the transverse Doppler effect, which is a real Doppler effect that occurs only at right angle. Both effects occur at right angles and in combination destroy the possibility to measure the transverse Doppler effect directly.....

The transverse Doppler actually doesn't exist, it is the "relativisic" Doppler Effect that occurs at all angles, which is caused by a slowing down of emission processes by motion in the gravitational field of the earth, which has nothing to do with some kind of time dilatation. There is no need to measure the transverse Doppler Effect at 90 degrees, but if you do it, it is difficult, because the emission angle permanently changes, if measured directly perpendicular to a moving light source.