Constraining the lifetime and opening angle of quasars using fluorescent
  Lyman $\alpha$ emission: the case of Q0420-388
  A toy model is developed to understand how the spatial distribution of
fluorescent emitters in the vicinity of bright quasars could be affected by the
geometry of the quasar bi-conical radiation field and by its lifetime. The
model is then applied to the distribution of high equivalent width Lyman
$\alpha$ emitters (with rest-frame equivalent widths above 100 A, threshold
used in e.g. Trainor & Steidel, 2013) identified in a deep narrow-band 36x36
arcmin$^2$ image centered on the luminous quasar Q0420-388. These emitters are
found to the edge of the field and show some evidence of an azimuthal asymmetry
on the sky of the type expected if the quasar is radiating in a bipolar cone.
If these sources are being fluorescently illuminated by the quasar, the two
most distant objects require a lifetime of at least 15 Myr for an opening angle
of 60 degrees or more, increasing to more than 40 Myr if the opening angle is
reduced to a minimum 30 degrees. However, few of the other expected signatures
of boosted fluorescence are seen at the current survey limits, e.g. a fall off
in Lyman $\alpha$ brightness, or equivalent width, with distance. Furthermore,
to have most of the Lyman $\alpha$ emission of the two distant sources to be
fluorescently boosted would require the quasar to have been significantly
brighter in the past. This suggests that these particular sources may not be
fluorescent, invalidating the above lifetime constraints. This would cast doubt
on the use of this relatively low equivalent width threshold and thus on the
lifetime analysis in Trainor & Steidel (2013).
