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We revisit the measurement of the polarization fraction, $P$, and themeasurement of the polarization angle of partially linearly-polarized gammarays using their conversion to $e^+e^-$ pairs in the field of a nucleus. Weshow that an inappropriate definition of the azimuthal angle, $\varphi$, usedto reference the orientation of the final state degrades the precision of themeasurement of $P$, by comparison to the optimal case where the bisector angleof the electron and of the positron momenta is used. We then focus on thelowest part of the energy spectrum, below $\approx 10\,$MeV, where a large partof the statistics lie for a cosmic source. We obtain the value of thepolarization asymmetry, $A$, of pair conversion at threshold and we show thatin the case where the correct expression is used for $\varphi$, the measuredvalue of $A$ tends to the limit.

$\gamma$-ray polarimetry with conversions to $e^+e^-$ pairs: polarization asymmetry and the way to measure it