Gas clumping factor ($C_e=⟨n_e^2⟩/⟨n_e{⟩}^2$)

The gas clumping factor measures unresolved inhomogeneity in electron number density within a chosen averaging region. Nonnegative variance gives $C_e\ge 1$. At a uniform temperature, thermal bremsstrahlung scales with $⟨n_e^2⟩$, whereas the thermal Sunyaev-Zeldovich effect scales with $⟨n_e⟩$. A homogeneous cluster distance from Sunyaev-Zeldovich and X-ray signals inference then gives $d_{A,\mathrm{inferred}}=d_{A,\mathrm{true}}/C_e$, for the same geometry and temperature. Correlated temperature variations require additional weighted averages.