Calculations with a full time-varying model are used to study changes in the height and density of the E-layer peak, caused by known changes in the neutral atmosphere. Agreement with mean observed values of N m E requires an increase of 10% in calculated ion densities, and an increase of 33% in the solar-maximum EUV model at l `150 A ร . At a ยฎxed site, changes with the solar zenith angle w agree well with the simple Chapman theory during most of the daylight hours. Simple modiยฎcations to the Chapman equations give improved accuracy near sunrise and sunset. When corrected for changes in w, model results for summer and equinox show a decrease in the peak density N m E at increasing latitudes. The overall change agrees well with experimental data, as summarised in the IRI model. Known changes in the neutral atmosphere also reproduce the increase in N m E in winter, at latitudes up to 308. The continuing increase at higher winter latitudes, in the IRI model, requires a major reduction in NO densities in winter. A suitable compromise is suggested. Equations ยฎtted to the model results then provide a simpler and better behaved replacement for the IRI equations. Calculations at night show that known sources of ionisation, largely from starlight, can produce observed peak densities using current chemistry. There is an appreciable change with latitude, as starlight production increases in the southern hemisphere. The improbably large solar cycle change built into the IRI model, at night, cannot be reproduced and is not found in recent data. A new, simpler model is suggested. Changes in zenith angle and atmospheric composition cause the peak height (h m E ) to vary between 105 and 120 km, as a function of time, latitude, season and solar ยฏux. These changes are approximated by simple equations that should be deยฎnitely preferable over the single, ยฎxed height used in the IRI models.