 Solution

The projected quasar-to-cloud separation isFor the usual transverse-separation estimate, take the cloud to lie at the same line-of-sight proper distance as the quasar. Since the cloud and galaxy are gravitationally bound, the local source-to-cloud path does not acquire the redshift factor $1+z=3$ that separates that system from us. Neglect internal relative Doppler shifts and attenuation. The observer in the cloud seesThis uses the quasar's isotropic intrinsic emission at the same rest frequency; anisotropic radiation or obscuration would modify it.

An angular separation only fixes $R_{\perp }$. If $s$ is a source-to-cloud displacement toward us, the true local distance is $r=\sqrt{R_{\perp }^2+s^2}$ and the general flux is $L_{\nu }/(4\pi r^2)$. The quoted transverse value is therefore an upper estimate at fixed projected separation. Equality of cosmological redshift and gravitational binding do not by themselves measure the small internal line-of-sight displacement; a peculiar velocity can also mask it.