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Saturday, October 3, 2015

Notes on Obliquity

It is usual, when discussing Milankovitch cycles, to dismiss any effect of obliquity on global, annual mean insolation based on the fact that the Earth is a sphere.  I did it myself in a comment on Skeptical Science.  As the Earth is an oblate spheroid, however, it is not strictly accurate.  In particular, on the equinoxes, the Earth presents its minimum aspect to the Sun, showing an eclipsed area of 1.2737 x 1014 m2 to the Sun.  As the Earth moves to the solstice (either winter or summer) it presents its maximum aspect, showing an eclipsed area of 1.2744 x 1014 m2.  That represents a difference of 0.05% in recieved sunlight, or approximately 0.12 W/m2.  That seasonal variation is much less than the 6.8% (16.2 W/m2) seasonal variation due to the eccentricity of the Earth's orbit.

As obliquity changes, the maximum eclipsed area (ie the insolation at the solstice) also changes, although the minimum eclipsed area (ie, insolation at the equinoxes) does not.  Just as the variation due to the Milankovitch cycle for eccentricity (~0.175%) is much smaller than the seasonal variation, so also is the variation due to the Milankovitch cycle in obliquity much smaller than the seasonal variation.  Specifically, it represents only a 0.01% variation in the solstice insolation.  As the equinoctial insolation does not change, the variation in global, annually averaged insolation due to changes in obliquity will be much less than 0.01%.

For perspective, these factors are also less than the difference of treating the Earth as a perfect oblate spheroid, or allowing for the additional interception of sunlight due to continents, mountains and, of course, the atmosphere.

(Edited from a comment on SkS)

Technical Notes (added July 22nd, 2016):

"Any planar cross-section passing through the center of an  forms an ellipse on its surface"

An oblate spheroid is an ellipsoid formed by rotating an ellipse around its minor axis.  It follows that a planar cross-section passing through the poles will be an ellipse with a major and minor axis equal to that of the oblate spheroid.  If we then tilt that cross-section, rotating it around the major axis, we will form a series of ellipses whose major axis is the same as that for the oblate spheroid and whose minor axis is given by the formula for the polar form of the generating ellipse (ie, the ellipse rotated to form the oblate spheroid) relative to the origin:

1)   r(θ) = ab/((bcos(θ))^2+(asin(θ))^2)^0.5

where  r(θ) is the minor axis of the ellipse formed by rotating the cross section by (90-θ) degrees from the vertical, and a is the major axis of the generating ellipse, while b is the minor axis of the generating ellipse.
Having determined the major and minor axes for the cross-section, its area is determined by the formula for the area of an ellipse:

2)  A = πab
where A is the area, a is the major axis, and b is the minor axis of the cross-section.

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