Interesting. In my old posts about the mathematics of the environment, which you’re now revisiting, I took the celestial mechanics underlying the Milankovitch as a black box, not having time to look into it. I naively assumed it was all worked out. But it’s tricky, especially over long timescales. The abstract of that paper you mentioned gives some clues as to why:

• J. Laskar, A. Fienga1, M. Gastinea, and H. Manche, La2010: A new orbital solution for the long-term motion of the Earth, *Astronomy and Astrophysics* **532** (2011), A89.

We present here a new solution for the astronomical computation of the orbital motion of the Earth spanning from 0 to −250 Myr. The main improvement with respect to our previous numerical solution La2004 is an improved adjustment of the parameters and initial conditions through a fit over 1 Myr to a special version of the highly accurate numerical ephemeris INPOP08 (Intégration Numérique Planétaire de l’Observatoire de Paris). The precession equations have also been entirely revised and are no longer averaged over the orbital motion of the Earth and Moon. This new orbital solution is now valid over more than 50 Myr in the past or into the future with proper phases of the eccentricity variations. Owing to the chaotic behavior, the precision of the solution decreases rapidly beyond this time span, and we discuss the behavior of various solutions beyond 50 Myr. For paleoclimate calibrations, we provide several different solutions that are all compatible with the most precise planetary ephemeris. We have thus reached the time where geological data are now required to discriminate between planetary orbital solutions beyond 50 Myr.

Luckily, 50 million years is enough to study the onset of the glacial cycles we’re experiencing now.

]]>One interesting aspect of eccentricity that isn’t apparent from the average insolation is the ratio between insolation at perihelion and insolation at aphelion. This insolation ratio is:

The ratio between naive greybody temperatures is the fourth root of this, which to highest order turns out to be simply:

So for , that’s something like a 17 degree difference in naive greybody temperatures between aphelion and perihelion.

But the phase of aphelion and perihelion relative to the seasons is also significant. Aphelion currently takes place in the northern summer, and the fact that there is more land in the northern hemisphere than the southern — with land having a much lower heat capacity than ocean — makes the average temperature of the Earth *higher* at aphelion than at perihelion. (Reference)

Interesting! So the maximum eccentricity is up from previous estimates, while the minimum is down.

]]>There’s a revised estimate of the historical eccentricity, which has now been reflected in the Wikipedia article:

The shape of the Earth’s orbit varies in time between nearly circular (with the lowest eccentricity of 0.000055) and mildly elliptical (highest eccentricity of 0.0679) with the mean eccentricity of 0.0019 as geometric or logarithmic mean.

They get this from:

• J. Laskar, A. Fienga1, M. Gastinea, and H. Manche, “La2010: A New Orbital Solution for the Long-term Motion of the Earth, *Astronomy and Astrophysics* **532** (2011), A89. PDF online.

• Tim van Beek, A quantum of warmth, Azimuth, 2 July 2011.

]]>Thanks! It would be great if you could add it to our list of Recommended Reading over on the Azimuth Project! We’ve already got a few books on climate modelling in that list.

]]>A very good introduction on climate modeling and simulation is Raymond Pierrehumbert’s book *Principles of Planetary Climate* from Cambridge Press.

I like his “hands on” philosophy: for example, there are many very interesting numerical projects in Python to give the intuitive feeling about this part of physics.

http://geosci.uchicago.edu/~rtp1/PrinciplesPlanetaryClimate/index.html

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