  We investigate mechanisms of wave-particle heating of ionospheric O$^{+}$
ions resulting from broadband extremely low frequency (BBELF) waves using
numerical test particle simulations that take into account ion-neutral
collisions, in order to explain observations from the Enhanced Polar Outflow
Probe (e-POP) satellite at low altitudes ($\sim$400 km)[Shen et al., 2018]. We
argue that in order to reproduce ion temperatures observed at e-POP altitudes,
the most effective ion heating mechanism is through cyclotron acceleration by
short-scale electrostatic ion cyclotron (EIC) waves with perpendicular
wavelengths $\lambda_{\perp}$$\leq$200 m. The interplay between finite
perpendicular wavelengths, wave amplitudes, and ion-neutral collision
frequencies collectively determine the ionospheric ion heating limit, which
begins to decrease sharply with decreasing altitude below approximately 500 km,
where the ratio $\frac{\nu_c}{f_{ci}}$ becomes larger than 10$^{-3}$, $\nu_c$
and $f_{ci}$ denoting the O$^+$--O collision frequency and ion cyclotron
frequency. We derive, both numerically and analytically, the ion gyroradius
limit from heating by an EIC wave at half the cyclotron frequency. The limit is
0.28$\lambda_{\perp}$. The ion gyroradius limit from an EIC wave can be
surpassed either through adding waves with different $\lambda_{\perp}$, or
through stochastic "breakout", meaning ions diffuse in energy beyond the
gyroradius limit due to stochastic heating from large-amplitude waves. Our
two-dimensional simulations indicate that small-scale ($<$1 km) Alfv\'en waves
cannot account for the observed ion heating through trapping or stochastic
heating.
