  We describe an electrothermal model for the turn-on dynamics of
superconducting nanowire single-photon detectors (SNSPDs). By extracting a
scaling law from a well-known electrothermal model of SNSPDs, we show that the
rise-time of the readout signal encodes the photon number as well as the length
of the nanowire with scaling $t_\text{rise}\propto \sqrt{\ell/n}$. We show that
these results hold regardless of the exact form of the thermal effects. This
explains how SNSPDs have inherent photon-number resolving capability. We
experimentally verify the photon number dependence by collecting waveforms for
different photon number, rescaling them according to our predicted relation,
and performing statistical analysis that shows that there is no statistical
significance between the rescaled curves. Additionally, we use our predicted
dependence of rise time on detector length to provide further insight to
previous theoretical work by other authors. By assuming a specific thermal
model, we predict that rise time will scale with bias current,
$t_\text{rise}\propto \sqrt{1/I_b}$. We fit this model to experimental data and
find that $t_\text{rise}\propto 1/(n^{0.52 \pm 0.03} ~I_b^{0.63 \pm 0.02})$,
which suggests further work is needed to better understand the bias current
dependence. This work gives new insights into the non-equilibrium dynamics of
thin superconducting films exposed to electromagnetic radiation.
