where is the 's . In general, the longitude of the sun will not be zero at the , but will be offset by an angle . for the , for example, occurs in January, while the is not until March. So it must be remembered that
We need to find the normal component of radiation at the north pole. But this flux will be simply
since the angular altitude of the from the horizon is given by . Using
and plugging (4) into (1) and (4) gives
Now, we are interested in finding the average of this flux over a full orbit. If the orbit is eccentric, the time required to travel an orbital distance is not constant, but related to dt according to
From (2),
which makes physical sense, since dt depends not on how the zero point for angular position is chosen, but on what the orbital distance is at the relevant position (the dependence). Equation (7) therefore becomes
To find the time-average flux, simply take
over the orbit. Note, however, that when L < 0 or (between autumnal equinox and vernal equinox), the north pole will be facing away from the sun and will therefore receive zero flux. The only nonzero contribution to the flux in (?) therefore occurs for .
