  The PHENIX collaboration presents first measurements of low-momentum
($0.4<p_T<3$ GeV/$c$) direct-photon yields from Au$+$Au collisions at
$\sqrt{s_{_{NN}}}$=39 and 62.4 GeV. For both beam energies the direct-photon
yields are substantially enhanced with respect to expectations from prompt
processes, similar to the yields observed in Au$+$Au collisions at
$\sqrt{s_{_{NN}}}$=200. Analyzing the photon yield as a function of the
experimental observable $dN_{\rm ch}/d\eta$ reveals that the low-momentum
($>$1\,GeV/$c$) direct-photon yield $dN_{\gamma}^{\rm dir}/d\eta$ is a smooth
function of $dN_{\rm ch}/d\eta$ and can be well described as proportional to
$(dN_{\rm ch}/d\eta)^\alpha$ with $\alpha{\approx}1.25$. This scaling behavior
holds for a wide range of beam energies at the Relativistic Heavy Ion Collider
and the Large Hadron Collider, for centrality selected samples, as well as for
different, $A$$+$$A$ collision systems. At a given beam energy the scaling also
holds for high $p_T$ ($>5$\,GeV/$c$) but when results from different collision
energies are compared, an additional $\sqrt{s_{_{NN}}}$-dependent
multiplicative factor is needed to describe the integrated-direct-photon yield.
