Last edited 10may03 by Matt Townsend
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illiTime : Exploring the Equation of Time
and the Analemmas of the Universe

Proposal

This program will explore some of the phenomenons behind the sun and its movement through the sky throughout the course of a day as well as throughout the course of a year. It will provide an animation of the sun and its position on the horizon at different times and explain its pattern of movement in the sky.

Abstract

IlliTime will examine and explain what is known as the Equation of Time. As many know, the sun does not always rise and set at 6:00, nor is it always at its highest point at noon. Instead, the sun seems to lag ahead or behind of this "expected" location throughout the course of the year. This amount, the difference in position the real sun is from what is known as the "mean sun", is given by the Equation of Time, and produces an affect known as the analemma. This program will observe the effects of this phenomenon, as well as explore similar phenomenons on other planets within our solar system and allow for simulation of custom planets; perhaps an orbiting comet or asteroid or even the moon will show to have an analemma of some kind.

A Brief Explanation

Due to the slightly elliptical orbit of the earth and its 23.5 degree tilt with respect to the sun, its journey through space throughout the course of a year causes a phenomenon known as the analemma. By neglecting these two factors, a perpendicular earth in a circular orbit would produce identical daily patterns of the sun in the sky; sunrise and sunset would forever occur at 6:00 and the sun would always be at zenith at noon. This, however, is obviously not the case. The combined effect of the tilt and the elliptical orbit cause the true position of the sun to stray as far as 15 minutes from the mean (expected) position of the sun. In addition, the 23.5 degree tilt causes the sun to travel along the horizon as the direction of the earth's axis facing the sun rotates. When these patterns are combined, the position of the sun at the same time tracked day by day throughout the course of the year will create the analemma shape in the sky. A similar phenomenon will create other analemmic patterns in the sky when the sun is viewed from the surfaces of other planets or any orbiting body. Patterns vary based on the eccentricity of the orbit, tilt of axis of the orbiting body, and length of day and year. By changing these parameters, a wide variety of analemmas can be observed.

The Equation of Time

The Elliptical Orbit:

As shown in the image, the earth's elliptical orbit causes it to travel faster at certain times of the year according to Kepler's laws, and as a result the earth is farther along in its orbit that it would be if it traveled at a constant speed. The earth will also fall behind at certain times throughout the year as its speed slows down. In the image, the angle a represents an angle of the "mean sun" measured from perihelion; that is, the angle the sun would have covered since perihelion if its orbit was circular. The angle v represents the angle of the "true sun" from its perihelion date. What we are interested in for calculating the Equation of Time is the difference in these angles. After all, the Equation of Time is a measure of the difference in position of the true sun from the mean sun as we observe it from the surface of the earth. After finding the difference between the two angles, we can find how many degrees in the sky the sun's position is offset from what we might expect. With a few simple calculations, this deviation can be converted to minutes, and we find that the sun strays from its mean position by almost 8 minutes at times. The resulting Equation of Time curve oscillates with a single period throughout the course of the year. Zero values occur at perihelion (Jan 3rd) and aphelion (July 5th), since the angles a and v are measured from perihelion. The above image shows the Equation of Time resulting from the earth's elliptical orbit.

Here is a more detailed mathematical explanation.

The Tilt of the Axis:

The above pictures show the effect of the tilted axis of the earth. This does not refer the movement of the sun across the horizon throughout the course of the year, specifically between the tropic of cancer and the tropic of capricorn. Although that movement does help to shape the analemma, the effects of the tilt as are relavent to the equation of time are different. Since the sun now revolves around the earth (although not really-the earth revolves around the sun, but its easier to understand for this circumstance) on a tilt, it will lag behind its mean position (that of the horizontal orbit) since it has to travel a farther distance, much like the hypotenuse of a right triangle is longer than its horizontal leg. This adds another offset to our equation of time formula that results from the tilt of the axis. We can again easily calculate how many minutes ahead or behind the sun is by finding the difference between the mean angle and the true angle, as is labeled in the second image. The resulting Equation of Time curve actually has two periods throughout the course of a single year; there are four zero values, one for each equinox (Mar 21 and Sep 21) and one for each solstice (Jun 21 and Dec 21). The graph above shows this curve resulting from the tilt of the axis.

Here is a more detailed mathematical explanation.

The Summation and Forming the Analemma:



Above are images of the analemma and how it is formed. When the two results from the equation of time, one from the elliptical orbit and one from the tilt of the axis, are added together, the final equation of time can be found. The graph of this summation is shown in the middle. The resulting curve gives shape to the pattern of the analemma. As mentioned before, the final factor contributing to this phenomenon is the movement of the sun from the tropic of cancer to the tropic of capricorn throughout the course of the year. This is also due to the tilt of the axis. When standing on the equator during the equinox, the sun will pass directly overhead, but as the earth approaches solstices in its orbit, the equator no longer represents the center of the earth that faces the sun. It might be at an equivalent lattitude of up to 23.5 degrees north or south, and the sun therefore sets at a different location on the horizon. The analemma is formed when its graph is reflected about the solstices, as shown on the right. The graphs for tilt and ellipse have been removed, and the summation has been reflected at July 21 and again at December 21. The resulting figure is the analemma.

Calculating the Orbit

The image to the right shows the orbit of the earth (blue) as it compares to a circular orbit (green). As is apparent, the difference is exremely slight, but does indeed exist. This difference is easier seen in planets with higher eccentricities, such as Mars and Pluto. Orbits are calculated from several key orbital elements; the eccentricity of the orbit, the semi-major axis, the inclination of the ecliptic, mean anomaly, argument of perihelion, and longitude of the ascending node. These are the orbital elements necessary to define the motion of any orbiting body around a center of gravity. This program demonstrates how analemmas can vary based on how elliptical the orbits are. For example, Jupiter has a fairly large eccentricity and an extremely small tilt, and so its analemma is dominated by its elliptical orbit. Neptune, however, has an eccentricity that is quite small, while its tilt is greater than that of the earth, and the analemma it forms is quite different. By observing the difference in the angle between the green and blue line as the simulation advances by day throughout the course of the year, it is clear where the curve resulting from the ellipse comes from.

Here is a more detailed mathematical explanation.

Analemmas and Orbits of Other Planets

By using these same methods of calculating the equation of time and the nature of the orbit, analemmas and traces can be simulated for most of the other 8 planets as well. The exceptions to this are Mercury and Venus. Mercury is tidally locked to the sun, and its period of rotation is exactly 2/3 its period of revolution. Since a year on Mercury is slightly longer than a day, Mercury does not produce an analemmic pattern that can be observed. Venus, although its year is longer, has an extremely slow rotation speed. To complicate matters even more, its spin is clockwise, backwards when compared to the counterclockwise motion of all other planets in the solar system. Excluding these two exceptions, analemmas can be observed from the surface of all other planets, and many other orbitting bodies for that matter. If the orbital elements are known for an orbiting asteroid or comet, analemmas can be calculated for these objects as well.

Here are some examples of other analemmic patterns and orbits.

Bibliography

www.analemma.com
Planet Reference
Orbit Calculation
Glossary of Orbital Elements