Why do the sensitivities of magnetometers and gravimiters have such strange units? It looks like the sensitivities of magnetometers and gravimeters are usually reported with the units of $\text{Tesla}/\sqrt{\text{Hz}}$ and $\text{Gal}/\sqrt{\text{Hz}}$, respectively (where "Gal" is a unit of acceleration).
Why is there the strange division by $\sqrt{\text{Hz}}$? I can't find an actual definition of the sensitivity of these sensors anywhere, so I don't know how to interpret those units.
 A: These figures represent the root mean square background measurement noise per unit bandwidth, in other words if you had an ideal bandpass filter placed after the instrument whose bandwidth is B [Hz] around some frequency at which those specified figures are given, say, $\sigma [T/\sqrt {Hz}]$ then the output will have some additive fluctuation whose standard deviation is $\sigma \sqrt{B}$
If your $\sigma$ is frequency varying, which is a common occurrence for no other reasons than that all instruments have $1/f$ noise, and you characterize the measuring instrument before it is sampled by a linear transfer function $H(f)$ normalized so that its peak is $max|H|=1$ then the total mean square fluctuation is
$$\sigma_M^2 = \int_0^\infty \sigma^2 (f) |H(f)|df$$
This $\sigma_M$ is then the rms fluctuation created by the instrument itself and is added to the actual measured sample. (Here we, of course, assume tacitly that the instrument noise is additive and Gaussian, as is indeed case usually.)
The equation for $\sigma_M$  is dimensionally consistent because if $\sigma (f)$ is measured in $[T/\sqrt{Hz}]$ then $\sigma^2(f)$ is measured in $[T^2/Hz]$, $H$ is dimensionless for it is normalized to $1$, so the dimension of $\sigma_M$ is then $[T]$ as it should be.
