Total irradiation during any time interval of the year using elliptic integrals

The calculation of the total solar energy received during a given interval of time over the year requires more attention than the calculation of the daily incoming solar radiation (daily insolation). It depends indeed upon whether the time interval is defined in terms of the true longitude or of the calendar date. The details of such a calculation based on elliptic integrals and on the sum of the daily irradiations are given in this paper. Numerical examples show the very high accuracy of both methods. The analytical expression and the spectral analysis of the long-term variations of such irradiation received during any time interval over the year show that they depend almost exclusively upon obliquity with a very small contribution of eccentricity, not at all upon precession. The eccentricity signal comes from the variation of the so-called solar constant through the variation of the mean distance from the Earth to the Sun. The precession signal is eliminated through the second law by Kepler. The correlation between total irradiation and obliquity reverses its sign at a specific latitude of the summer hemisphere which is a function of the selected time interval. At this latitude the obliquity signal disappears and only a pure eccentricity signal is left in the total irradiation, but with a very weak amplitude. Amplitude of the continuous wavelet transforrm shows that the amplitude variation of both the daily irradiance (mostly precession) and of the total irradiation (obliquity) vanishes around the Mid-Pleistocene Transition. (C) 2010 Elsevier Ltd. All rights reserved.