# Why is the transverse Doppler shift “difficult to measure”?

I'm currently reading a book on optics, and have encountered a curious section:

$$\nu = \nu'\sqrt{1-\frac{u^2}{c^2}} = \nu'\left(1-\frac{u^2}{2c^2}+\ldots\right)$$

This is the formula for the transverse Doppler shift, giving the frequency change when the relative motion is at right angles to the direction of observation. The transverse Doppler shift is a second-order effect and is therefore very difficult to measure. It has been verified by using the Mossbauer effect with gamma radiation from radioactive atoms.

What about this specific effect makes it difficult to measure? I understand that the text says it is because it is a second-order effect, but it's not clear to me why that makes a correction term so much more difficult to observe.

Is there a good elucidation on the reasons behind this?

Because you need to have sufficiently high $$\frac{u}{c}$$ value in the experiment so that the term $$\left(\frac{u}{c}\right)^2$$ will be significant and the corresponding change in frequency is detectable.
For eg. If you have $$\frac{u}{c} = 0.001$$ which is literally having particles moving at a velocity of $$300000$$ m/s. Then the change you detect is just $$10^{-6}$$ only so it is difficult to observe when compared to first order $$10^{-3}$$.