We present results from a study of the time lags between changes in the energy flow into the polar regions and the response of the thermosphere to the heating. Measurements of the neutral density from the CHAMP and GRACE missions are used, along with calculations of the total Poynting flux entering the poles. During two major geomagnetic storms in 2003 these data show increased densities are first seen on the dayside edge of the auroral ovals after a surge in the energy input. At lower latitudes the densities reach their peak values on the dayside earlier than on the night side. A puzzling response seen in the CHAMP measurements during the November 2003 storm was that the density at a fixed location near the “Harang discontinuity’ remained at unusually low levels during three sequential orbit passes, while elsewhere the density increased. The entire database of measurements from the CHAMP and GRACE missions were used to derive maps of the density time lags across the globe. The maps show a large gradient between short and long time delays between $60^{\circ}$ and $30^{\circ}$ geographic latitude. They confirm the findings from the two storm periods, that near the equator the density on the dayside responds earlier than on the nightside. The time lags are longest near 18 – 20 h local time. The time lag maps could be applied to improve the accuracy of empirical thermosphere models, and developers of numerical models may find these results useful for comparisons with their calculations.