Past exam of the astrophysics course of the University of Oxford / 2015 / 8 / d / Solution

For a bound-cloud light echo, the relevant observed delay is the excess source–cloud–observer path over the direct source–observer path, with the interval dilated by cosmological expansion. For a cloud displaced by $s$ toward us,The formula assumes source and cloud separation is small compared with their cosmological distance, so their distant observer directions are effectively parallel. For the transverse configuration $s=0$,The local quasar-to-cloud travel time is about $2.86\times 10^5$ yr; the factor three is needed for the interval measured by us. A foreground cloud shortens the echo delay and a background cloud lengthens it. A unique value beyond this transverse estimate needs the otherwise unspecified depth, even if the cloud's intrinsic response is instantaneous.