What is the Path of the Sun?

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What Is the Path of the Sun?
Teacher's Notes

Focus questions

How does the position of the sun at a given time each day change throughout the year?
What causes this change?

Essential materials

Indoor project:

a room with a window that faces the sun
a small mirror, about 2 cm (3/4 inch) in diameter
glue or rubber cement
a safe ladder
bits of paper and tape
an accurate clock that measures to the nearest second

Outdoor project:

a flagpole, telephone pole, or other similar object that casts a shadow onto a paved area around midday
all-weather paint
accurate clock that measures to the nearest second
if no flagpole or telephone pole is available:
- a flat, clear paved area, about 3 m by 3 m (10 ft by 10 ft), on which you can make paint marks
- a rigid stake or rod, at least 1.25 m (4 ft) long
- a heavy hammer, such as a sledgehammer

Main ideas and background information

If somehow it were possible to mark the position of the sun in the sky at exactly the same time of the day for an entire year, the resulting shape would be an elongated figure eight. This distinctive invisible path of the sun is called an analemma.

The analemma is determined mainly by two factors: the tilt of the Earth’s axis and the shape of the Earth’s orbit. Let’s look at each of these separately.

Although we know that the orbit of the earth is elliptical (oval), imagine for the moment that it is a perfect circle, and that the rotational axis is tilted as we know it to be. Now think about the effect of the tilt on how the sun appears to us through the course of a year. There are two aspects to consider.
- One is that the height of the sun in the sky at a given time varies throughout the year. For example, in winter the sun appears lower in the sky at noon in the Northern Hemisphere than it does in summer.
- The other aspect is that during the course of a year, there is a maximum and a minimum height that the sun appears in the sky at a given time of day.
The tilt of the Earth also affects the left-right position of the sun in the sky at a given time of day. This is a bit more difficult to visualize. Think of it this way: When you look at a globe, you can see that the farther the longitudinal lines are from the equator, the closer they are together.
- During the summer and winter solstices (June 21 and December 21), the noon sun is directly over the Tropic of Cancer and the Tropic of Capricorn, where the longitudinal lines are closer together than at the equator.
- During the vernal and autumnal equinoxes (March 21 and September 21), the noon sun is directly over the equator, where the longitudinal lines are farther apart. Local time is directly related to local longitude.
- As a result, the speed of the sun seems to increase in a left-right or east-west direction two times each year, during the spring and fall. Conversely, the east-west speed of the sun seems to decrease during the summer and winter.
The tilt of the Earth causes both these apparent east-west speed changes and the north-south motion of the sun during the year. Taken together, these motions are responsible for the basic figure eight shape of the analemma.

If the orbit of the Earth truly were a perfect circle, these up-down and left-right effects caused by the tilt of the axis would produce a symmetrical analemma such as the one pictured here. The solstices would be at the ends of the figure eight, and the equinoxes would be at the crossover point. For more detailed information, see www.analemma.com .

Now let’s look at the second factor: the actual orbit of the Earth. The orbit’s slightly elliptical shape means that Earth travels faster through space when it is closer to the sun between early October and early March. When Earth reaches its perihelion (point of closest distance to the sun) in early January, it is orbiting at its top speed. Between early March and early October, when it reaches its aphelion (point of farthest distance from the sun) in early July, the Earth is traveling at its slowest speed in its orbit.

From our perspective from our tilted Earth, the location of the sun appears to shift to the left or right with respect to local time. In the course of a year, this produces a maximum speed and a minimum speed of the position of the sun in a left-right direction at a given time each day. For more detailed information, see www.analemma.com .

Currently, the Earth reaches its perihelion between January 2 and January 5, nearly two weeks after the winter solstice, which is usually around December 21. Similarly, the Earth is at its aphelion between July 3 and July 7, about two weeks after the summer solstice, which is usually around June 21.

The maximum and minimum speeds of the slightly elliptical orbit do not change the basic shape of the analemma. However, they do create an asymmetrical analemma, with the winter loop somewhat larger than the summer loop. As a result, the points representing the equinoxes are slightly offset from the cross of the analemma.

Here is a summary of the impact of the tilt of the axis and the shape of the orbit on the dimensions of the analemma.

	Tilt of axis	Shape of orbit
Analemma width	X	X
Analemma length	X

The time of day selected for viewing the sun determines the slope of the analemma. An analemma developed early or late in the day will be tilted, while one created from midday observations will be perpendicular to the horizon.

The analemma explains why the date with the shortest period of daylight does not fall on the date with the latest sunrise or the earliest sunset. For more detailed information, see "Why the earliest sunset, latest sunrise, and shortest day of the year occur on different dates"
by John Holtz .

Procedural tips

For the indoor project, the small mirror must be flat rather than concave or convex. Flat dental mirrors work well for this activity.
Be sure to inform any maintenance staff about the mirror or other analemma artifacts so that nothing is accidentally moved or discarded.
If you are using shadows, be certain to mark the same part of the shadow consistently.
Accurate timing is essential. Make sure your clock displays seconds. Have students make their marks consistently from day to day to the nearest second if possible. It helps to have your students poised for action at least a minute before the selected time for recording.
The sun in the sky appears to move a distance equal to its diameter in about two minutes. This means that if you are a few seconds early or late making the mark, it is not a serious problem. However, a minute or two will make a noticeable difference.
It is not necessary to make a mark each day, so do not worry if you have a number of cloudy days.

Safety considerations

Do not allow your students to look directly at the sun at any time. This can result in permanent eye damage.
Check for sturdiness and stability any ladders, chairs, tables, etc., that are used for extra height. Handrails would be a plus.
Do not allow any horseplay whatsoever, especially when anyone is on a ladder or chair.
Position students near the ladder or chair to act as "spotters" in case the person who is placing the mark gets off balance.
Be sure that each student steps down off the ladder or chair slowly and safely.

Discussion

Why does the analemma have a figure eight shape?
The analemma shows the path of the apparent position of the sun as seen at the same time each day while the Earth travels one time around its orbit. The tilt of the Earth’s axis gives the analemma its up-down length. From December to June, the height of the sun in the sky changes between a maximum and a minimum height. The tilt of the Earth also creates two maximum and two minimum left-right speeds of the sun’s apparent position. These four left-right speed changes and the two up-down position changes result in a figure eight shape.
What would the analemma look like if the Earth’s orbit were a perfect circle?
The analemma would be caused only by the tilt of the Earth’s rotational axis. The result would be a symmetrical figure eight, with the solstices at the ends and the equinoxes at the crossover point.
Why does the actual analemma not cross on the dates of the equinoxes?
The effect of the elliptical orbit of the Earth distorts the symmetrical analemma of the axis tilt, creating the long and short loops seen in the actual analemma.
What makes analemmas different from one another?
The time of day selected for marking the position of the sun determines the slope of the analemma. The latitude of the observation location also has an impact on the analemma slope. The analemma size depends on the distance from the mirror to the wall or ceiling, or the distance from the object to its shadow.
What do you think an analemma would look like as seen from the hemisphere opposite of where you live?
The small loop of the analemma would point in a northward direction when seen from either hemisphere. The slope of the analemma will depend on the time of day selected. The tracing of the analemma during the year would go around the figure eight in the direction opposite that of the other hemisphere.
How is the analemma related to a sundial?
The two are closely related. The sundial uses the blade (called the gnomon) to cast a straight shadow over an arc of numbers to tell roughly the hour of the day. The analemma could be used with a sundial to give a more accurate time. Additional information is available at
Analemma Sundial Applet
Design of the Richard D. Swensen Sundial

Assessment

Are students able to determine how the position of the sun at a given time each day changes throughout the year? (The position traces out a path shaped like a figure eight. The path is called an analemma.)

Are students able to explain what causes this change? (The analemma is caused by the combined effects of Earth’s tilted axis of rotation and of Earth’s slightly elliptical orbit around the sun.)

Extensions and further investigations

Have students repeat the project at several different times of day. This will create a display of several analemmas developing simultaneously.
The tilt of the Earth’s axis is about 23.5 degrees. Ask the students to predict what would happen to the analemma in various locations around the world if the tilt measured 45 degrees. What about a tilt of 10 degrees?
Encourage your students to create a “weekend analemma” by selecting something in or near their home that casts a convenient shadow and marking it at the same time each Saturday or Sunday. Suggest that they get their family members involved in the project as well.
Suggest that students experiment with digital or video cameras for this project. For example, if a student uses a digital camera to photograph a shadow several times during a year from exactly the same position and orientation and then “plays” the photographs as a video, the shadow would trace out the analemma shape.
Challenge students to design a similar project for another planet in the solar system, predicting the expected results and giving their rationale.

Career connections

Astronomy
Photography
Sundial design

Correlations with Standards

United States: This activity correlates with portions of NSES Content Standard A, Science as Inquiry, and Content Standard G, History and Nature of Science, Grades 5-8 and 9-12, and with the following additional standards:

Grades 5-8
Earth and Space Science: D3

Britain: This activity correlates with the English National Curriculum standard Sc1, Science Enquiry, and the following additional standards:

KS2, Sc4: 4b, c, BoS: 1c, d; KS3, Sc4: 4a, b, c, BoS: 1d, e, f; KS4, Sc4: 4a, BoS: 1d, e, f

Glossary/vocabulary

elliptical
perihelion
aphelion
equinox
solstice
analemma
gnomon