  Photons are a penetrating probe of the hot medium formed in heavy-ion
collisions, but they are emitted from all collision stages. At photon energies
below 2-3 GeV, the measured photon spectra are approximately exponential and
can be characterized by their inverse logarithmic slope, often called
"effective temperature" $T_\mathrm{eff}$. Modelling the evolution of the
radiating medium hydrodynamically, we analyze the factors controlling the value
of $T_\mathrm{eff}$ and how it is related to the evolving true temperature $T$
of the fireball. We find that at RHIC and LHC energies most photons are emitted
from fireball regions with $T{\,\sim\,}T_\mathrm{c}$ near the quark-hadron
phase transition, but that their effective temperature is significantly
enhanced by strong radial flow. Although a very hot, high pressure early
collision stage is required for generating this radial flow, we demonstrate
that the experimentally measured large effective photon temperatures
$T_\mathrm{eff}{\,>\,}T_\mathrm{c}$, taken alone, do not prove that any
electromagnetic radiation was actually emitted from regions with true
temperatures well above $T_\mathrm{c}$. We explore tools that can help to
provide additional evidence for the relative weight of photon emission from the
early quark-gluon and late hadronic phases. We find that the recently measured
centrality dependence of the total thermal photon yield requires a larger
contribution from late emission than presently encoded in our hydrodynamic
model.
