How much bigger is $a$ than $b$ when we write $a \gg b$? When a condition of some physical quantity $a$ and $b$ is presented, say $a \gg b$, how much bigger do we say $a$ has to be than $b$? $10$ times? $1000$ times? $10^6$ times?
Some context:
I am trying to use adiabatic theory to derive the energy spectrum for a particle confined in a ring of radius $b$ in the x-y plane, when a Dirac monopole is slowly moved at a constant velocity $v$ from $z = -\infty$ to $z = \infty$. The condition for applicability of the adiabatic theorem is that the time $T$ that it takes for a varying parameter to change appreciably is much larger than $\frac{\hbar}{\Delta E}$, where $\Delta E$ denotes some typical energy level difference between eigenstates of the Hamiltonian. My problem is, I can't directly quantify an appreciable change, say by $10 \%$ in $z$, since we can write $z = z_0 + vt$ but $z_0$ is $-\infty$. So, I was thinking of taking $z$ to  be essentially infinitely far away from the ring when $|z| \gg b$, the radius of the ring. This would give me a value for $z_0$. But I am not sure how much bigger to make $z$ than $b$, hence my question.
 A: The usual treatment for the approximation is to consider the ratio $1\gg \frac{b}{a}$ and proceed to Taylor expand the function of your interest around some point, such that the displacements (perturbations) are written in terms of this ratio. Then you can decide where to truncate your series expansion and estimate what is the error you are making by bounding the truncated terms. 
My suggestion for your case would be then to try to write exact expressions first, if possible. Then do as I say above. It is possible that you may need to translate the adiabatic condition into some other variables that appear directly in your formulas.
EDIT: A common example would be the expansion of a term of the form:
$$\left(1-\frac{b}{a}\right)^{-n} =\sum_{k=0}^\infty \binom{n+k-1}{k}x^k = 1 + n\frac{b}{a} + \mathcal{o}\left(\left(\frac{b}{a}\right)^2\right) \approx 1 + n\frac{b}{a}$$
(You can read about the little-o notation  in Wikipedia which has the precise definition). In the last step, the series was truncated to first order (all higher order terms are ignored). However, you can prove that the error you are commiting can be bounded by the next derivative (after truncation), so in our case by the second derivative, and by the displacement. This series is centered at 0, so the displacement would be just |x|. For the details, see Taylor's Theorem. I hope the example works to understand the principle. 
In the end is up to you to say how much precision you need, to state at which point your conclusions are valid.
