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Contents

Idea

The Milankovitch cycles are periodic or quasiperiodic changes in the parameters of the Earth’s orbit and tilt, which in turn affect the climate. The three major types of Milankovitch cycle are:

• changes in the eccentricity of the Earth’s orbit:

• changes in the obliquity, or tilt of the Earth’s axis:

• precession, meaning changes in the direction of the Earth’s axis relative to the fixed stars:

These changes do not effect the overall annual amount of solar radiation hitting the Earth, but they affect the strength of the seasons in different ways at different latitudes. It is widely believed that they are partially responsible for the glacial cycles. However, the details of how this occurs are complex and poorly understood, as the following graph makes clear. Some form of amplification, e.g. stochastic resonance, may be involved.

History

A brief history of Milankovitch cycles and their effect on glaciation is given in:

• Michel Crucifix, Glacial-interglacial cycles, in Springer Encyclopaedia of Snow, Ice and Glaciers.

Let us quote a bit:

Details

There are three major forms of Milankovitch cycle:

Eccentricity: The Earth’s orbit is an ellipse, and the eccentricity of this ellipse says how far it is from being circular. But the Earth’s orbit slowly changes shape: it varies from being nearly circular (eccentricity of 0.005) to being more strongly elliptical (eccentricity of 0.058), with a mean eccentricity of 0.028. There are several periodic components to these variations. The strongest occurs with a period of 413,000 years, and changes the eccentricity by ±0.012. Two other components have periods of 95,000 and 125,000 years.

The eccentricity affects the percentage difference in incoming solar radiation between the perihelion, when the Earth is closest to the Sun, and aphelion, when it is farthest from the Sun. This works as follows. The percentage difference between the Earth’s distance from the Sun at perihelion and aphelion is twice the eccentricity, and the percentage change in incoming solar radiation is about twice that. The first fact follows from the definition of eccentricity, while the second follows from differentiating the inverse-square relationship between brightness and distance.

Right now the eccentricity is 0.0167, or 1.67%. Thus, the distance from the Earth to Sun varies 3.34% over the course of a year. This in turn gives an annual variation in incoming solar radiation of about 6.68%. Note that this is not the cause of the seasons, which arise due to the Earth’s tilt, and occur at different times in the northern and southern hemispheres.

Obliquity: The angle of the Earth’s axial tilt with respect to the plane of its orbit, called the obliquity, varies between 22.1° and 24.5° in a roughly periodic way, with a period of 41,000 years. When the obliquity is high, the strength of seasonal variations is stronger.

Right now the obliquity is 23.44°, roughly halfway between its extreme values. It is decreasing, and will reach its minimum value around the year 10,000 CE.

Precession: The slow turning in the direction of the Earth’s axis of rotation relative to the fixed stars, called precession, has a period of roughly 26,000 years. As precession occurs, the seasons drift in and out of phase with the perihelion and aphelion of the Earth’s orbit.

Right now the perihelion occurs during the southern hemisphere’s summer, while the aphelion is reached during the southern winter. This tends to make the southern hemisphere seasons more extreme than the northern hemisphere seasons.

The gradual precession of the Earth is not due to the same physical mechanism as the wobbling of the top. That wobbling does occur, but it has a period of only 427 days. The 26,000-year precession is due to tidal interactions between the Earth, Sun and Moon. For details, see:

• John Baez, The wobbling of the Earth and other curiosities.

Glacial cycles

The relation between the Milankovitch cycles and glacial cycles is complex, fascinating, but also poorly understood and controversial. One can see why here:

• The blue curve on top shows the obliquity, $\epsilon$.

• The green curve is the eccentricity, $e$.

• The purple curve is $\sin \varpi$, where $\varpi$ is the longitude of the perihelion, that is, the angle between the vernal equinox and perihelion with the Sun as the angle vertex. This goes from 0 to $2 \pi$ as precession takes place.

• The red curve is the precession index $e \sin \varpi$, which plays an important role (see Insolation for details).

• The black curve summarizes the combined effects of all Milankovitch cycles: $\overline{Q}^{day}$ is the daily-averaged incoming solar radiation at the top of the atmosphere on the day of the summer solstice at 65° north latitude, measured in watts/meter². This says how much solar energy the Earth gets on midsummer’s day at this latitude. To see how this is calculated from the previous quantities, see Insolation.

• The benthic foram data comes from Lorraine E. Lisiecki and M. E. Raymo, A Pliocene-Pleistocene stack of 57 globally distributed benthic δ¹⁸O records, Paleoceanography 20 (2005), PA1003. Benthic forams are little shells found in ancient sediments on the sea floor. Much of the change in oxygen-18 in these shells is thought to be caused by sequestration of the more common lighter isotope of oxygen, oxygen-16, in large continental ice sheets.

• The Vostok ice core data comes J. R. Petit et al., Vostok ice core data for 420,000 years, IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series #2001-076, NOAA/NGDC Paleoclimatology Program. It uses isotopic analysis from an ice core sample taken from Vostok, Antarctica: the temperature is inferred from the concentration of deuterium in the ice.

The above graph was taken from Wikimedia Commons. The Milankovitch cycles were computed using Fortran programs available here:

• NASA, Determination of the Earth’s orbital parameters.

At this website you can create your own table of the eccentricity, obliquity and longitude of perhelion for selected years by entering the minimum year, maximum year, and yearly increment.

Stochastic resonance

There have been interesting attempts to argue that stochastic resonance is important in amplifying the small effects of Milankovitch cycles to create glacial cycles. Work along these lines began with Benzi, Patera, Sutera, and Vulpiani:

• R. Benzi, A. Sutera, and A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A 14 (1981), L453.

• R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, Stochastic resonance in climatic change, Tellus 34 (1982), 10–16.

For a review of this line of work see:

• R. Benzi, Stochastic resonance: from climate to biology, Nonlin. Processes Geophys. 17 (2010), 431–441.

Also see:

• A. V. Kislov, Astronomical forcing and mathematical theory of glacial-interglacial cycles, Climate of the Past Discussions 5 (2009), 327–340.

• Michel Crucifix, Jonathan Rougier: On the use of simple dynamical systems for climate predictions: A Bayesian prediction of the next glacial inception. Summary here on the Azimuth Project.

For a fascinating but highly mathematical treatment using energy balance models see:

• Peter Imkeller, Energy balance models - viewed from stochastic dynamics, Progress in Probability 49 (2001), 213–???.

For more, see Stochastic resonance.

The effect of changes in eccentricity

Changes in obliquity clearly have very little effect on the global, annual average insolation, since the Earth is nearly spherical. It is more tricky to determine how changes in the eccentricity of the Earth’s orbit affect this quantity. Let us work this out in detail, following a calculation presented by Greg Egan on the Azimuth blog. While the result is surely not new, Egan’s approach makes nice use of the fact that both gravity and solar radiation obey an inverse-square law!

Here is his calculation:

The angular velocity $\frac{d \theta}{d t} = \frac{J}{m r^2}$, where $J$ is the constant orbital angular momentum of the planet and $m$ is its mass, so if the radiant energy delivered per unit time to the planet is $\frac{d U}{d t} = \frac{C}{r^2}$ for some constant $C$, the energy delivered per unit of angular progress around the orbit is

So, the total energy delivered in one period will be $U=\frac{2\pi C m}{J}$.

How can we relate the orbital angular momentum $J$ to the shape of the orbit? If you equate the total energy of the planet, kinetic $\frac{1}{2}m v^2$ plus potential $-\frac{G M m}{r}$, at its aphelion $r_1$ and perihelion $r_2$, and use $J$ to get the velocity in the kinetic energy term from its distance, $v=\frac{J}{m r}$, when we solve for $J$ we get:

where $a=\frac{1}{2} (r_1+r_2)$ is the semi-major axis of the orbit and $b=\sqrt{r_1 r_2}$ is the semi-minor axis. But we can also relate $J$ to the period of the orbit, $T$, by integrating the rate at which orbital area is swept out by the planet, $\frac{1}{2} r^2 \frac{d \theta}{d t} = \frac{J}{2 m}$ over one orbit. Since the area of an ellipse is $\pi a b$, this gives us:

Equating these two expressions for $J$ shows that the period is:

So the period depends only on the semi-major axis; for a fixed value of $a$, it’s independent of the eccentricity.

If we agree to hold the orbital period $T$, and hence the semi-major axis $a$, constant and only vary the eccentricity of the orbit, we have:

Expressing the semi-minor axis in terms of the semi-major axis and the eccentricity, $b^2 = a^2 (1-e^2)$, we get:

So to second order in $e$, we have:

The expressions simplify if we consider average rate of energy delivery over an orbit, which makes all the grungy constants related to gravitational dynamics go away:

or to second order in $e$:

We can now work out how much the actual changes in the Earth’s orbit affect the amount of solar radiation it gets! In what follows we assume that the semi-major axis is indeed held constant while eccentricity changes; from the work of Laskar (see below) this is approximately correct. According to Wikipedia:

The shape of the Earth’s orbit varies in time between nearly circular (low eccentricity of 0.005) and mildly elliptical (high eccentricity of 0.058) with the mean eccentricity of 0.028.

The total energy the Earth gets each year from solar radiation is proportional to

where $e$ is the eccentricity. When the eccentricity is at its lowest value, $e = 0.005$, we get

When the eccentricity is at its highest value, $e = 0.058$, we get

So, the change is about

In other words, a change of merely 0.167%.

That’s very small And the effect on the Earth’s temperature would naively be even less!

Naively, we can treat the Earth as a greybody. Since the temperature of a greybody is proportional to the fourth root of the power it receives, a 0.167% change in solar energy received per year corresponds to a percentage change in temperature roughly one fourth as big. That’s a 0.042% change in temperature. If we imagine starting with an Earth like ours, with an average temperature of roughly 290 kelvin, that’s a change of just 0.12 kelvin!

The upshot seems to be this: in a naive model without any amplifying effects, changes in the eccentricity of the Earth’s orbit would cause temperature changes of just 0.12 °C!

This is much less than the roughly 5 °C change we see between glacial and interglacial periods. So, if changes in eccentricity are important in glacial cycles, we have some explaining to do. Possible explanations include season-dependent phenomena and climate feedback effects.

References

News articles and overviews

This news item surveys 17 papers in an issue of Paleoceanography entitled ‘Milankovitch climate cycles through geologic time’:

• Richard Kerr, Milankovitch climate cycles through the ages, Science 235 (1987), 973-974.

This news item provides a quick summary of John Imbrie’s work on the SPECMAP project, which uses spectral analysis to look for effects of the Milankovitch cycles on different aspects of our climate:

• Richard Kerr, Mapping orbital effects on climate, Science 234 (1986), 283-284.

• Ice Rhythms: Core Reveals a Plethora of Climate Cycles. A discussion of how ice core samples aid in determining past changes in climate, further illustrating the concept of the Milankovitch cycle.

• Possible Forcing of Global Temperature by the Oceanic Tides. A discussion of fluctuation in global temperatures and the trends due to solar output. (?)

• Nils-Axner Morner, Terrestrial, solar and galactic origin of the Earth’s geophysical variables, 1984. A breakdown of changes to the earth that have resulted from galactic, solar and terrestrial changes.

• A New View on The Driving Mechanism of Milankovitch Glaciation Cycles. A discussion of how glacial cycles could be strongly influenced by one of the Milankovitch cycles in particular.

• Milanovitch Climate Reinforcements. A discussion of how the Milankovitch cycles may play a lesser role in global climate change, serving only to reinforce and enhance current patterns.

• The Ice Chronicles: The Quest to Understand Global Climate Change - The study of glacial changes from ice cores to form a record of past climate changes.

• J Hansen and M Sato, Draft paper Paleoclimate Implications for Human-Made Climate Change, intended for Milankovic conference, 2011.

Technical papers

Hays, Imbrie and Shackleton 1976

• J. D. Hays, J. Imbrie, and N. J. Shackleton, Variations in the earth’s orbit: pacemaker of the Ice Ages, Science 194 (1976), 1121-1132.

This was a major analysis that applied Fourier analysis to climate proxies. From several hundred sediment cores studied stratigraphically by the CLIMAP project, they selected two (RC11-120 and E49-18) from the Indian ocean and measured:

1) $\delta^{18}O$, the oxygen isotopic composition of planktonic foraminifera,

2) Ts, an estimate of summer sea-surface temperatures at the core site, derived from a statistical analysis of radiolarian assemblages,

3) percentage of Cycladophora davisiana, the relative abundance of a radiolarian species not used in the estimation of Ts.

Identical samples were analyzed for the three variables at 10-cm intervals through each core.

Here are their findings for RC11-120:

and for E49-18:

After some processing described in the paper, here are the power spectra for these data:

As you can see, there seem to be peaks at frequences of 100 ka, 42 ka, 23 ka, and 19 ka.

Berger 1989

This piece of work reviews evidence for the Milankovitch explanation of glacial cycles, at least up to 1989:

• A. Berger, Pleistocene climatic variability at astronomical frequencies, Quaternary International

2 (1989), 1-14.

Berger cites the aforementioned paper of Hays, Imbrie and Shackleton as important in finding empirical evidence for long-term climate cycles. He writes:

Laskar et al

Laskar seems to be an authority on celestial mechanics. Relevant papers of his include:

1. J. Laskar, The chaotic motion of the solar system. A numerical estimate of the chaotic zones, Icarus 88 (1990), 266-291.

2. J. Laskar, F. Joutel and F. Boudin, Orbital, precessional and insolation quantities for the Earth from -20 Myr to + 10Myr, Astronomy and Astrophysics 270 (1993), 522-533.

3. J. Laskar, M. Gastineau, F. Joutel, P. Robutel, B. Levrard, and A.C.M. Correia, A long term numerical solution for the insolation quantities of Earth. Astronomy and Astrophysics 428 (2004), 261-285.

In (2) section 4.1, they state that for the purpose of computing insolation, they neglect the secular variations of the semi major axis. There is a diagram for the variation of the semi-major axis in (2) (fig. 11) which seems to justify this. In (3) section 7, they state that an early version of this fact was known to Laplace and Lagrange!

In these papers they describe a rather complete computation of orbit elements, obliquity and precession for several tens of millions of years. There is some uncertainties, for instance due to the tides on Earth. In (2) 7.2 it is taken into account that the magnitude of the tides might change during an ice age. This is further discussed in [3] 4.2, but I’m not entirely convinced that they really really know by how much. They do seem pretty convinced that their calculations are correct from -20 My to present. If you try to extend the calculations to earlier times, you will eventually be up against chaos. In (3) section 11 they state that it is hopeless to try to get to the Mesozoic, before - 65My.

The best place to start reading these papers is probably (3), especially because of the interesting diagrams and tables.

Bennett 1990

Here is a curious paper on Milankovitch cycles and evolutionary biology:

• K. D. Bennett, Milankovitch cycles and their effects on species in ecological evolutionary time, Paleobiology 16 (1990), 11–21. A look at the biological aspects of the effects the Milankovitch cycles has on our planet.

Gallée et al 1991-1992

• H. Gallée, J. P. van Ypersele, T. Fichefet, I. Marsiat, C. Tricot and A. Berger, Simulation of the last glacial cycle by a coupled, sectorially averaged climate–ice sheet model. Part I: The climate model, Journal of Geophysical Research 96 (1991), 13139–13161.

• H. Gallée, J. P. van Ypersele, T. Fichefet, I. Marsiat, C. Tricot and A. Berger, Simulation of the last glacial cycle by a coupled, sectorially averaged climate–ice sheet model. Part II: response to insolation and CO₂ variation, Journal of Geophysical Research 97 (1992), 15713–15740.

These papers present a model that simulates the last glacial cycle starting from insolation data over the last 122 kiloyears. Ice albedo feedback plays a crucial role in amplifying the Milankovich cycles. The importance of many other effects is studied, with many interesting conclusions. The effect of changing CO₂ concentrations is neglected except in Section 6 of the second paper, and even here the feedback cycle involving temperature and CO₂ concentration is not discussed.

There are many interesting figures. Here is one of the most basic: a graph of June insolation as a function of latitude for the last 130,000 years:

Huybers 2009

• P. Huybers, Pleistocene glacial variability as a chaotic response to obliquity forcing, Clim. Past Discuss. 5 (2009), 237-250.

In this paper, Huybers proposes the following: The eccentricity period does not matter, that’s a red herring. Throw it away, don’t think about it. It’s all about tilt (obliquity). The glaciation 40ky period indeed corresponds to the dominant obliquity period. Sometimes, this period gets doubled or tripled, giving periods of 80 ky or 120 ky. These two cases he subsumes under the common name $\sim100$ ky.

The problems he wants to solve is: why does doubling and tripling occur, and why do we get sequences of single periods (as in early Pleistocene) , and sequences of repeated periods (as in late Pleistocene and Holocene). That is, following a single period, we will most likely get a single period, and following a $\sim 100$ky period, we will most likely get a $\sim100$ ky period.

As a preliminary and as supporting evidence he makes one strange observation about the $\delta^{18}O$ data. The curve $\gamma$ describing this, which presumably also measures the total amount of ice on the planet, has the following strange property: The derivative $\gamma\prime$ is negatively correlated to $\gamma$, with a time lag of 9 ky. The long time lag is curious, and he does suggest several possible physical explanations for this correlation, without chosing his favourite. It seems (though it’s not stated absolutely clearly in the article), that this correlation is mainly valid in the ablation phase, that is during de-glaciation.

An interesting fact is that the correlation persists through the change from 40ky to $\sim100$ ky periods. This seems to imply that the physics of ablation is the same in both cases. I think that Huybers takes this as evidence that the forcing is also the same. Since eccentricity can’t be blamed when the period is 40ky, eccentricity can’t be guilty even when the period is 100ky.

Based on this observation, Huybers writes down a time dependent difference equation, whose time dependence codifies a periodic forcing, with period 40 ky. He states that the difference equation has chaotic behaviour, but also that there exists a periodic orbit with period 40ky. Close to this periodic orbit, there are solutions with periods 80ky. These solutions alternate between 40ky of mild glaciation and 40 ky of severe glaciation. There are also cycles of period 120 ky (I don’t know if they are also supposed to be close to the basic periodic orbit) , passing through 40ky of interglacial, 40 ky of mild glacial and finally 40 ky of full glacial. So it seems that we can produce sequences of cycles close to these periodic orbits which reproduce the geological record. Unfortunately I did not entirely understand the mathematics of this part of the article.

The system is chaotic, so we should expect random shifts between the various periodic orbits. There is no particular external reason for the shift from 40ky to 100ky.

Miscellaneous online references

• http://aa.usno.navy.mil/faq/docs/seasons_orbit.html. A detailed description on the seasons’ relationship to the Milankovitch cycle.

• http://geography.about.com/library/weekly/aa121498.htm and http://earthobservatory.nasa.gov/Library/Giants/Milankovitch/. Biography of Milankovitch and details on his theory of climate change.

• http://journalofscience.wlu.edu/archive/Spring2000/GlobalWarming.html. A discussion of global warming and climate change as related to several causes, including the Milankovitch cycle.

• http://lwf.ncdc.noaa.gov/oa/climate/globalwarming.html#Q1. A large FAQ on climate change and global warming. Discusses the Milakovitch cycle as well as several related topics such as solar output and sun spots.

• http://www.ngdc.noaa.gov/stp/SOLAR/solarda3.html. A data base including climate change models related to solar changes and the Earth’s orbit.

• http://www.physicalgeography.net/fundamentals/6i.html. Earth-Sun relationships in terms of insolation.

• http://www.whoi.edu/institutes/occi/currenttopics/climatechange_wef.html#n1. Animations and global cooling ideas.

• Global Climate Change Student Guide. A guide to climate change, with an extensive section devoted to the Milankovitch Theory and cycles.

• http://www.nature.com/nsu/020819/020819-9.html. Nature editorial raising the possibility that the next glacial might be prevented by global warming.

Books

Let’s list books chronologically and comment on them. Here are some:

• Milutin Milanković‘s original book is available in German under the title Kanon der Erdebestrahlung und Seine Anwendung auf das Eiszeitenproblem. It was also translated into English in 1969 under the title Canon of Insolation of the Ice-Age Problem by the Israel Program for Scientific Translations, and published by the U.S. Department of Commerce and the National Science Foundation, Washington, D.C..

• J. Imbrie and K. P. Imbrie, Ice Ages, Solving the Mystery, Enslow Publishers, Short Hills, New Jersey, 1979.

• A. Berger, J. Imbrie, J. Hays, G. Kukla, and B. Saltzman, editors, Milankovitch and Climate, D. Reidel Publ. Company, Dordrecht, Netherlands, 1984.