## Friday, January 17, 2014

### Milankovitch Cycles 3

This is the third part of a series on the celestial cycles that periodically influence Earth's climate. For the first part, see here. For the second, see here.

In the first two parts of this post, two types of precession and their combined effect on the Earth's climate were explored. However, the Earth's orbit, as well as rotating over time, also varies in eccentricity. Currently, the eccentricity of Earth's orbit is about 0.017. However, due to slight perturbations from the other planets, the elongation of the ellipse that is Earth's orbit changes over time, going as low as 0.005 (nearly a perfect circle) and as high as 0.058. The mean value of the cycle is 0.028.

Unlike the variations discussed thus far, eccentricity variation does not have a simple cycle, but is rather a sum of component cycles, with all of the planets adding a contribution, some greater than others. Since these components are of different periods, simply evaluating the period by multiplying yields an exact period longer than the age of the Solar System. Thus one must use an "approximate" period that yields a cyclic variation, but within a certain error margin. Such a approximation yields a period of about 100,000 years, but with error margins that vary considerably.

The above diagram is an analysis of the periods of the eccentricity variations in the Earth's orbit. The x-axis has lengths of periods in thousands of years, organized logarithmically. For example, the value of 3 on the x-axis corresponds to a period of 23*1000 = 8000 years. The y-axis indicates "how closely" the eccentricity variation has a component with a period of a given length. For example, as indicated, there is a peak in the curve labeled "413", meaning that it occurs at 413,000 years. The peak indicates that there is a significant component of the variation with a period of this length.

Overall, the eccentricity variation has the effect of accentuating or dampening the effects of axial and apsidal precession. Before we delve into the climatological effects more fully, one more variation must be considered: axial tilt variation.

In addition, to the Earth's axis precessing in a circular motion, the obliquity of the axis, or the angle between the line through the center of the Earth perpendicular to the plane of the Earth's orbit and the line of the Earth's poles, changes in an approximately periodic manner. This process is also called nutation.

A diagram illustrating the obliquity of the Earth's axis. However, the tilt shown in the figure, which is currently about 23.4°, is not constant. It too changes with time, with a period of about 41,000 years, varying from 22.1° to 24.5° over its cycle.

The image shows the highest and lowest obliquities that Earth's axis experiences during its cycle. The current obliquity is near the midpoint of the two end values and is decreasing towards 22.1°. However, the amplitude of the variation of the obliquity over its period is not constant. The high and low values shown in the above diagram represent the most extreme variation, but some 41,000 year periods have seen variation of less than half a degree.

The Moon, which exerts gravitational forces on the Earth, acts to stabilize the variation in the position of the Earth's axis. It has been estimated that, without the Moon, the Earth's obliquity could vary enormously, on the scale of tens of degrees over its period. The influence that the Moon has on stabilizing the Earth's axis is quite important in keeping climate patterns stable. Without it, the Earth's climate could vary chaotically, with formerly polar regions becoming tropical and vice versa within tens of thousands of years which, though still a long period, would be devastating to delicate ecosystems and biodiversity.

Both of these variations have great effects on climate, the most significant of which are the amplitudes of seasonal variation. Near the highest eccentricity over the period of variation in its orbit, the Earth receives over 20% more sunlight (per unit area) at perihelion than aphelion. Due to the dampening effect of the insulation of Earth's atmosphere, the planet's heat content does not vary to this degree. However, seasonal variation would increase significantly, making these phases of the eccentricity cycle more susceptible to ice ages, as is reflected in paleoclimatological records.

Thus far, we have been treating the properties of the orbit of the Earth and all its effects on climate as if it were lying exactly in the plane of the Solar System. However, the orbit is also inclined from the plane, and this inclination varies. Before discussing the climatological effects of this variance, we must pin down precisely what the "plane of the Solar System" is.

On Earth, we typically consider the positions of bodies in the sky relative to the plane of the ecliptic, which is the plane of Earth's orbit (the ecliptic itself is the "intersection" of this plane with the celestial sphere). However, abandoning this geocentric view, we consider what is called the invariable plane. To find the invariable plane, one must find the barycenter, or center of mass, of the Solar System. The Solar System as a whole (neglecting any internal interactions) behaves as if it is a single object with a mass equivalent to that of the Solar System at its barycenter. The average rotational properties of all bodies of the Solar System provide the angular momentum vector for this ersatz representation of the system, and the invariable plane is defined as the plane passing through the barycenter perpendicular to this vector.

The angular momentum vector for an object is perpendicular to the plane of rotation, as in the top above, where L is the direction of the vector. Similarly, the angular momentum of the imaginary object representing the Solar System is perpendicular to its (again imaginary) plane of rotation.

The Sun, being on top of the center of gravity of the Solar System, does not contribute significantly to its overall angular momentum. The biggest contributions come from the gas giants, particularly Jupiter. The angle between the plane of Earth's orbit and this invariable plane is about 1.57°, which varies with a period of about 100,000 years. The places where the Earth's orbit crosses the invariable plane "going up" and "going down" are called the ascending node and descending node, respectively, and these presently occur on July 9 and January 9, respectively. Note that these nodes can be defined with respect to any plane, but we are taking it to be the invariable plane in this instance.

The above figure is a summary of some other orbital parameters of a celestial body, including the ascending and descending nodes and their longitudes, i.e., their angle from a reference direction in the invariable plane. The inclination of Earth's orbit and the position of these nodes have subtle and varied effects on climate; the positions of the nodes effect where the Earth passes through meteor showers. These, in turn, create atmospheric dust, especially in the polar regions, that form high-altitude clouds, as water vapor clings to the dust particles. These clouds are called noctilucent clouds, and a possible correlation between their abundance and ice ages has been observed in geologic records.

Milankovitch cycles help to explain at least some of the cycles of cooling and warming that the Earth experiences. However, some important disclaimers must be made. First, some feedback mechanisms help to accentuate or dampen any climatological changes brought about by orbital or rotational properties. For example, in a warming period, the polar ice caps shrink, releasing some trapped carbon dioxide that causes further warming. Therefore, even changes that are started by the Milankovitch cycles cannot be solely attributed to the cosmos, as they are but the beginning of a chain reaction here on Earth.

Second, there are many climage changes that cannot be explained through Milankovitch cycles. Large meteoroid impacts, volcanic eruptions, and other Earth-based phenomena could change or even counteract any effect of the orbit on Earth's climate, and often work in much shorter time periods.

Despite these shortcomings, the Milankovitch cycles have explained much about past climate variation on Earth, and may play a significant role in the future. The complicated web of cause and effect between the cycles and various periods of Earth's geologic history is not yet fully elucidated, but many significant correlations have been observed that promise to help us understand our planet's history.