= Central and averaged Compton y parameters of a uniform sphere
{title2=$y(b)=y(0)\sqrt{1-b^2/R^2},\quad\bar y=2y(0)/3$}
In a uniform spherical <plasma> of <radius> $R$, <electron> <number density> $n_e$ and <temperature> $T$, the <thermal Sunyaev-Zeldovich effect> gives $y(0)=2\sigma_TRn_ek_BT/(m_ec^2)$ along the central ray. A chord at impact parameter $b$ is shorter by $\sqrt{1-b^2/R^2}$. Averaging $y(b)$ over the projected disk with area element $2\pi b\,db$ gives $\bar y=2y(0)/3$. Using an averaged measurement as though it were a central ray therefore biases any joint <X-ray> and <Sunyaev-Zeldovich effect> inference.
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