Why we typically see no deeper into an atmosphere for an optical depth of 1?

Question

In An Introduction to Modern Astrophysics (p.243), Carroll and Ostlie say that for a optical depth $\tau = 1$ the intensity will decline by a faction of $e^{-1}$.

$I_\lambda = I_{\lambda,0}e^{-\tau}$

They also say that as result, we typically see no deeper into an atmosphere for an optical depth $\tau \approx 1$.

However, using the formula above, the intensity is still $\approx 0.37$ of the initial intensity, but the way I understand the authors the intensity should be null for $\tau \approx 1$.

Answer 1

When the authors say $\tau\approx 1$ they are giving a order of magnitude estimate i.e. $\tau$ is nearer to $1$ than it is to $10$ or $0.1$. On a log scale this would mean $\tau \lt \sqrt{10}$ and indeed $\exp(-\sqrt{10}) \approx 0.04$, which is close to zero.