= Solution
In the <thermal Sunyaev-Zeldovich effect>, hot intracluster <electrons> transfer <energy> to <cosmic microwave background> <photons> by <inverse Compton scattering>. Their thermal <energy> is much larger than the <CMB> <photon> <energy>; in the nonrelativistic limit the mean fractional <photon> <energy> gain per scattering is about $4k_BT/(m_ec^2)$. Multiplication by the incremental <Thomson scattering> <optical depth> $d\tau=\sigma_Tn_e\,dx$ explains why $y=\int[k_BT/(m_ec^2)]\,d\tau$ measures the electron <pressure> integrated along the line of sight. The result is a spectral distortion, with a low-<frequency> decrement and a high-<frequency> increment, rather than a positive additional emission component at every <frequency>.
For a uniform sphere, a line of sight at impact parameter $b$ has path length $2\sqrt{R^2-b^2}$. Therefore
$$
\boxed{y(b)=\frac{2\sigma_T}{m_ec^2}n_ek_BT\sqrt{R^2-b^2},\qquad 0\le b\le R.}
$$
On the central line of sight this reduces to the requested expression
$$
\boxed{y(0)=\frac{2\sigma_T}{m_ec^2}Rn_ek_BT.}
$$
The central-ray qualification matters: the average over the projected disk is $\bar y=(2/3)y(0)$, while the tangent ray $b=R$ has $y(R)=0$. Thus the printed formula does not hold for an arbitrary sightline. Part <3/f> uses the specified $y$ as this central value, as required by the preceding formula.
Back to article page
