Orbital Mechanics ("Milankovich Cycles" Part 3)

Figure 3-1: “Periodicity” refers to the time required for the orbit to change from the most circular to the most elliptical. (Indiana.edu)

Figure 3-1: “Periodicity” refers to the time required for the orbit to change from the most circular to the most elliptical. (Indiana.edu)

This is third article in a series by RCA Member David Horne. His first article discussed Milankovich Orbital Cycles and Earth’s Ice Ages. The second discussed Ice Age erratics and Earth's Orbital Cycles. This installment will complete the discussion of Earth’s orbital cycles. We can then turn to the Milankovich Theory, its origins and predictions.

There are three basic cycles involved in Earth’s orbit around the sun: eccentricity, orbital precession, and orbital inclination.


The Earth’s orbit around the Sun is in the shape of an ellipse with the Sun closer to one end of the ellipse than the other. The elliptical shape of the orbit is termed its eccentricity, a measure of the deviation of the Earth’s orbit from a perfect circle. 

The eccentricity of the Earth's orbit changes over time. This occurs because Earth is influenced in its orbit by the gravitational fields of other planets, primarily Jupiter and Saturn. The diagram above, Figure 3-1, greatly exaggerates the degree of ellipse for illustration; in fact it is not very great. 

Earth’s orbit varies from nearly circular to mildly elliptical. Currently we are near the most circular orbit and are moving towards the nearly circular portion of our cycle. If the Earth’s orbit is a perfect circle, then the eccentricity would have a value of 0. An eccentricity value of 1 would indicate a parabola. Over the years, Earth’s eccentricity has varied between 0.005 and 0.0607. Today the eccentricity of Earth’s orbit is approximately 0.0167.

In percentage terms today, this difference results in only about a 3% change in Earth’s distance from the sun at the farthest point in its orbit (aphelion) and the closest point in its orbit (perihelion). The shape of the orbit by itself, today, causes only about a 6% difference between the amount of solar energy received in January vs. July. But, at the most elliptical, the difference between the solar energy received in January vs. July (as a result only of this orbital feature) is on the order of 20% to 30%! 

The eccentricity cycle is not stable, but subject to cyclical variations composed primarily of two smaller cycles of 95,000 and 113,000 years. These two cycles overlap or “beat” together in a primary 100,000-year cycle, with a secondary cycle of 413,000 years. As these cycles overlap, the cyclical periods of reinforcement and partial cancellation are termed “beat frequencies.”

Beat frequencies are overlapping cycles which occur when two cycles are out of phase with one another in time. In orbital features, when the variations are cycling at different time periods, there will be periods of time when two or more of them occur at the same time. These periodic combinations of reinforcement and cancellation have their own beat frequency when two or more cycles periodically “beat” together. They don’t combine in to one cycle, they just occur coincidently at overlapping points in time on a regular basis. It is this overlap that is termed the beat frequency. An example would be the pulsing sound of a twin propeller airliner when one propeller is rotating at a different rate than the other. The low pulsing sound of one propeller blade periodically occurs at the same time as the other creating a louder and annoying pulsing sound. The frequency of the pulse, how often per unit of time the pulse occurs, is the beat frequency.

Figure 3-2: How a beat frequency works (Hyperphysics concepts)

The diagram, figure 3-2, shows how this works with a simple two cycle system. The first chart represents two separate, similar, but out of phase frequencies. Note that they regularly come together and then separate out to maximum distance in a repeating pattern, a cycle. The second chart is a graph of this regular repeating cycle of the two coming together then separating out, a graph of the beat frequency.

In a simple case, dealing say, with two audio tones or radio waves of different frequencies, the beat frequency can be calculated with precision. (As in the diagram above.) Note also that the amplitude of the signal is unaffected. But, in a complex natural system such as Earth’s orbital cycles (orbital frequencies) which are not so precise in their timing or frequency, the resulting beat frequencies are not so simple. It can be very messy.

The degree of eccentricity has only a small impact on incoming solar radiation at the top of the atmosphere. Its primary impact on the amount of solar radiation striking Earth, comes from its modulating effect caused by the axial tilt and axial precession cycles in spreading the radiation received over a larger area on Earth. The tilt determines the timing and intensity of the seasons for each hemisphere. Eccentricity cycles will moderate this effect by reinforcing or minimizing the effect of the tilt on insolation. When the orbit of the Earth is highly elliptical one hemisphere will have hotter summers and colder winters, corresponding to a larger average yearly insolation. When the orbit is circular, seasonal differences tend to decrease.

Figure 3-3: Aspidal Precession of the Moon (Wikkicommon)

Orbital (Aspidal) Precession

Not only does the shape of the orbit change, but the orbit itself revolves around the Sun. This feature of the orbit is termed apsidal precession, or precession of the ellipse. Think of Earth’s orbit around the Sun as the motion of a hula hoop around its user, with the user being the sun. (The moon’s orbit behaves in a similar fashion, first noticed by the ancient Greek astronomer Hipparchus.) If one were to put a piece of tape at a point on the hula hoop/orbit, it would take approximately 112,000 years for the hoop to complete one revolution or circle. And remember that while this is going on, the shape of the orbit (the shape of the hula hoop) is also changing. The diagram of the Moon’s apsidal precession, figure 3-3, may help to visualize the motion of this feature.    

Figure 3-4: Changes in the Orbital Plane of the Earth (Diagram by Donald Etz)

Orbital Inclination

There is one final aspect of Earth’s orbit about the Sun of relatively more recent discovery: orbital inclination. As amateur astronomers are aware, it refers to the plane described by the orbit of the planets, including Earth, around the sun. The plane of the orbit is influenced and set primarily by Jupiter. The orbits of almost all the planets lie within this plane, the plane of the ecliptic. But the orbital plane of the individual planets, including Earth, wander up and down a bit. That is, the inclination of the plane of Earth’s orbit cycles up and down, and this “wandering”, sometimes termed orbital inclination, goes through cycles of approximately 100,000 years. It is currently 1.58°, and is decreasing. Its last maximum, about 30,000 years ago, was about 2.6°, and it is expected to decrease to a minimum of about 0.8° in about 20,000 years. It affects the angle at which the Sun’s rays strike the Earth. The following diagram, figure 3-4, greatly exaggerates the degree of variation for purposes of illustration. 

Orbital Mechanics Summary

These then are the features of Earth’s orbital and rotational mechanics. It is helpful to keep several points in mind. First, the first two orbital features affect the intensity of incoming solar radiation at the top of the atmosphere (insolation) received by the Earth at various points in its orbit. The closer to the radiation source, the more radiation is received. (Recall that intensity of radiation varies as the square of the distance from the source. A friend once paced off the distance between our tents at a star party to reduce the intensity of my snoring by 75%.)

Second, the last two rotational features (and to some extent orbital inclination) affect the distribution of the incoming solar radiation at various latitudes. That is, rotational tilt and precession affect the amount of insolation received depending on the latitude: the higher the latitude the greater the angle so the greater the effect. This is so because the latitude determines the angle at which the sun’s rays strike Earth at that point. A larger angle of tilt works to spread the sunlight received over a larger area of Earth at higher latitudes.

FIgure 3.5: Effects of precession on the seasons (Wikimedia Commons: Krishnavedala

Third, the direction of Earth’s axis determines the point in Earth’s orbit the solstices and equinoxes/seasons occur. Precession will cause a season to occur at a slightly different place in Earth’s orbit each year. And this occurs while Earth’s orbit itself is changing as we reviewed. These effects work together to shift where each season occurs, and the point at which it occurs in Earth’s orbit around the Sun. Currently, Summer occurs at the furthest distance on Earth’s slightly elliptical orbit around the sun. The solstice and the equinox always fall on the same day of the same month each year, but they are set by a point on the orbit which changes slightly each year. So, we must add a little bit of time periodically to keep the solstice and equinox on the same days and at the same point on a changing orbit. This is termed the “Precession of the Seasons”, and it means, for example, that December 21, 1 B.C.E. occurred in what is roughly our March. The following diagram, figure 3-5, illustrates the changing relationship over time between the axial tilt, season, and distance from the Sun. And of course, while this is going on, the shape of the orbit is changing, the orbit itself is revolving like a hula hoop and the plane of the orbit is wandering up and down.

Next month we will turn to the Milankovich Theory, its development and predictions. 

Thanks to our Science Editor, Dr. Katherine Kornei, for her assistance with this article.