Validity of the approximated emissivity given by Eq. (B.3). This figure shows the ratio of (X^1/2 + 2^11/12X^1/6)² to X [or equivalently, of the emissivity (B.1) to its approximation (B.3)] as a function of magnetic field strength B and dimensionless temperature Θ_e, fixing sin θ = 1 and ν = 230 GHz (and neglecting redshift effects). The solid white curves are level sets of ratios of 1.5, 2 and 3. The dashed red curves are level sets of the X parameter corresponding to X = 10 and X = 100.