We investigate the relationship between a galaxy cluster’s hydrostatic equilibrium state, the entropy profile, K, of the intracluster gas, and the system’s non-thermal pressure (NTP), within an analytic model of cluster structures. When NTP is neglected from the cluster’s hydrostatic state, we find that the gas’ logarithmic entropy slope,
$k\equiv \mathrm{d}\ln K/\mathrm{d}\ln r$, converges at large halocentric radius, r, to a value that is systematically higher than the value
$k\simeq1.1$ that is found in observations and simulations. By applying a constraint on these ‘pristine equilibrium’ slopes,
$k_\mathrm{eq}$, we are able to predict the required NTP that must be introduced into the hydrostatic state of the cluster. We solve for the fraction,
$\mathcal{F}\equiv p_\mathrm{nt}/p$, of NTP,
$p_\mathrm{nt}$, to total pressure, p, of the cluster, and we find
$\mathcal{F}(r)$ to be an increasing function of halocentric radius, r, that can be parameterised by its value in the cluster’s core,
$\mathcal{F}_0$, with this prediction able to be fit to the functional form proposed in numerical simulations. The minimum NTP fraction, as the solution with zero NTP in the core,
$\mathcal{F}_0=0$, we find to be in excellent agreement with the mean NTP predicted in non-radiative simulations, beyond halocentric radii of
$r\gtrsim0.7r_{500}$, and in tension with observational constraints derived at similar radii. For this minimum NTP profile, we predict
$\mathcal{F}\simeq0.20$ at
$r_{500}$, and
$\mathcal{F}\simeq0.34$ at
$2r_{500}$; this amount of NTP leads to a hydrostatic bias of
$b\simeq0.12$ in the cluster mass
$M_{500}$ when measured within
$r_{500}$. Our results suggest that the NTP of galaxy clusters contributes a significant amount to their hydrostatic state near the virial radius and must be accounted for when estimating the cluster’s halo mass using hydrostatic equilibrium approaches.