Materials Included In Kit

Additional Materials Required

Prelab Preparation

1. Locate a room that faces the south side of the building and receives sun at the local noon time throughout the year.
2. The Sun’s Altitude Sheet Master should be copied and cut out before class. The individual Sun’s Altitude Sheets may be taped to the gnomon paddles before beginning the activity.

Safety Precautions

Although this activity is considered nonhazardous, please follow all normal laboratory safety guidelines. Never look directly at the Sun.

Disposal

The materials in this activity may be saved for future use.

Lab Hints

• Extra Sun’s Altitude Templates have been provided in case students would like to make their own gnomon paddles.
• All of the master sheets may be copied and given to students so they can take their own data.
• Methods for calculating local noon time may be found online. Reminder: All local noon times must be at Standard Time.
• It is suggested that this activity be performed by the instructor before student use to assure class understanding.
• The Sun’s Altitude Sheet should be flush with the gnomon stick before it is taped in place.
• This activity should be started on the Fall Equinox (September 20th or 21st). This will allow for a baseline reading for the Analemma graph and allow students to record data until class ends in the early summer.
• The gnomon paddle should be level before taking data. A level is mounted on the gnomon paddle to ensure that the paddle is not tilted.
• If an adequate position in the classroom is not found, select an area outdoors or in a hallway.
• The gnomon paddle should be placed in the same position for each reading.
• Taking data at local noon time will ensure accurate representation of the Sun’s altitude at that time each day. Sunrise and sunset data may be found on the Internet or in a local newspaper the day of each reading. If this is not possible due to classroom schedules, the data recording time may be altered.
• Readings should be taken, at the very least, on the 1st and 15th of each month.
• If it is cloudy on any day when data is to be recorded, the reading should be taken on the next clear day.
• The figure-8 form will become evident as the data is recorded throughout the year.
• If desired, this activity may be more inquiry based by not giving the “scale” for the Sun’s altitude readings in steps 11–12 and by having students plot data by figuring out the declination angles (see next Lab Hint).
• The declination angle (where the Sun is directly overhead or 90° to the horizon) is calculated by the following method.
  1. If your latitude is 40 degrees north and the Sun’s angle for June 1 is 72 degrees, then the corresponding angle is 18 degrees (90 – 72 = 18). Subtract the 18 degrees from your latitude of 40 degrees north. The result is 22 degrees. This is the declination angle for June 1. Place a dot at the intersection of 22 and +2.10 on your chart. Follow this procedure for the entire year.
  2. Note: During the months of March to September you subtract the corresponding angle from your latitude. During the months of September to March you add the corresponding angle to your latitude (see Figure 9).
     {12787_Hints_Figure_9}
• Values present on the Equation of Time Chart Master are averages and may be in error by 10 to 15 seconds in December and January of certain years.
• Classroom data may also be taken on a classroom-sized Analemma graph sheet, if desired.
• The Sun’s altitude angle may also be measured directly with a protractor using the following procedure:
  1. Tape a plain sheet of paper to the gnomon paddle.
  2. Measure the height of the shadow stick. This will be side a.
  3. Measure and mark the length of the shadow. This will be side b.
  4. Side c will be the distance from the top of the shadow stick to the end of the shadow (see Figure 10).
     {12787_Hints_Figure_10}
  5. The length of side c can be found mathematically using the Pythagorean Theorem (Equation 1):
     {12787_Hints_Equation_1}
     {12787_Hints_Equation_8}
  6. The angle of the Sun’s altitude (in degrees) may be found directly using a protractor or by using Figure 11 and Equation 2.
     {12787_Hints_Figure_11}
     {12787_Hints_Equation_2}
     {12787_Hints_Equation_9}
• The first scientifically sound approximation of the Earth’s circumference based on direct measurement was made by Eratosthenes about 200 B.C. At Syene, Egypt, a place near the 23.5 degrees latitude, at the time of the summer solstice (June) 21, it could be observed the Sun’s noon rays were nearly perpendicular to the Earth’s surface, so as to shine upon the floor of a deep vertical well. On the same solstice date at Alexandria, located approximately on the same longitude as Syene, the Sun’s noon rays were blocked by an obelisk creating a shadow measured to make an angle of 7 degrees with respect with the perpendicular. Assuming the rays to be parallel, it is evident that the Earth’s surface must be curved in an arc of 7 degrees. The distance between Syene and Alexandria was estimated to be about 5000 stadia. Multiplying by 50, Eratosthenes estimated the Earth’s circumference to be about 250,000 stadia. In today’s terms, that is about 46,250 kilometers or 26,660 miles. Eratosthenes’ method was sound in principle and has since been applied with great precision to determine the Earth’s form and dimensions.
• A rough estimation of the circumference of the Earth may be made using the difference between the internal Earth angles from two different sites. The internal Earth angle is the compliment angle to the Sun’s angle. The internal Earth angles can be observed directly on the gnomon paddle. In order to calculate the Earth’s circumference, the internal Earth angle must be measured at exactly the same time in two locations. The locations should be directly North and South of one another and the mileage between the two should be known. Once the variables have been obtained the circumference of the Earth may be calculated as shown in Equation 3:
  {12787_Hints_Equation_3}
  For example, if Town A is directly 690 miles north of Town B and the measured Earth’s Internal angle for Town A was 50 and Town B was 40, the circumference of the Earth would be calculated in Equation 4.
  {12787_Hints_Equation_4}
• The circumference of the Earth may also be calculated as follows:
  1. Find the Sun’s altitude in degrees along your latitude at local noon time.
  2. Calculate the number of degrees away from an observer’s latitude where the Sun’s rays are directly overhead at local noon, subtract your latitude degrees form the compliment angle.
     
     Use the following method from September 21 until March 22. Example: Sun angle is 35 degrees; compliment angle is 55 degrees. Therefore, if you are 40 degrees north latitude, the Sun would be directly overhead at the 15 degrees south latitude (55 – 40 = 15).
     
     Use the following method from March 22 until September 21. Since the Sun’s direct rays are in the northern hemisphere, reverse the equation (i.e., observer’s latitude minus compliment angle). Therefore, if you are 40 degrees north latitude, the Sun angle would be directly overhead at the 10 degrees north latitude (40 – 30 = 10).
     
     Given
     • The Sun’s angle is 35 degrees
     • Earth’s interior angle is 55 degrees (compliment angle)
     • Sun’s declination is 15 degrees south latitude
     Facts
     • 360 degrees in a circle
     • 1 degree of latitude equals 69 miles
     Solve
     • d = the distance between the observer’s latitude and where the Sun’s rays are directly overhead
     • Multiply the number of miles of 1 degree latitude (69) per angle i (55°) times
       Example: 55° x 69 miles = 3795 miles
     Equation
     {12787_Hints_Equation_5}
     Substitute
     {12787_Hints_Equation_6}
     Earth’s Circumference

     C = 24,840 miles

Teacher Tips

• The study of the analemma in the classroom will allow students to:
  1. Gain a deeper understanding of the models of Earth, time, and space.
  2. Process abstract ideas about the Earth’s rotation and orbit around the Sun.
  3. Learn more about the terrestrial day and the solar year.
• Materials and instructions to have students create their own sundials may be found in Flinn Scientific’s kit Sun–Earth Motion (Catalog No. AP6625).
• A completed sample Analemma Graph (from 40 °N Latitude) is presented:
  {12787_Tips_Figure_12}

Sample Data

Latitude ___40º N___

Answers to Questions

1. Define the following terms: Analemma, ecliptic, equation of time, declination angle and gnomon.

   Analemma—is the path that the sun makes in the sky over a year’s time.
   Ecliptic—the path the Sun and the planets follow across the sky.
   Equation of Time—The difference between clock time and solar time.
   Declination angle—the angle that the Sun makes with the plane of the ecliptic throughout the year.
   Gnomon—a “shadow stick” that was used as the earliest form of a sundial.

2. Why does the sun appear in a different location in the sky each day?

   Two factors contribute to why the Sun appears to be in a different location in the sky each day—the orbit of Earth around the Sun and the tilt of the Earth’s axis. When the Earth is traveling along its elliptical, or oval-shaped orbit, and is closer to the Sun (winter in the Northern hemisphere) it speeds up slightly. In the summer in the Northern hemisphere, the Earth is farther away from the Sun and it slows down slightly. The tilt of the Earth’s axis also causes the Sun to appear early or late. The Sun and the planets follow a path across the sky called the ecliptic. The view of the ecliptic from Earth is tilted due to the Earth’s 23.5-degree tilted axis. The effect of this tilt produces an apparent slowing down and speeding up of the Sun as it moves along the ecliptic.

3. What pattern was observed on the Analemma chart?

   A figure 8 pattern was observed.

4. Describe possible sources of error that may have occurred during this activity.

   The Sun was not “out” on the correct data reading days, the gnomon paddle was moved from previous readings, the gnomon was not level, the data readings were not taken at the exact local noon time, etc.

References

Special thanks to Ed Colangelo, New Jersey, for providing the idea for this activity to Flinn Scientific.

Introduction

What is an analemma and how does it relate to the position of the Sun in the sky? Perform the following activity to answer these questions.

Concepts

• Analemma

• Sun movement
• Earth movement
• Day length
• Seasons

Background

Have you ever seen the figure-8 pattern (see Figure 1) on a globe and wondered what it was? This figure-8 pattern is known as an analemma. An analemma is the path that the Sun makes in the sky over a year’s time. If the position of the Sun is recorded at the same time every day it would be seen that the Sun not only varies in an upward and downward direction, but also leftward and rightward positions at certain times of the year (see Figure 1).

Two factors contribute to why the Sun appears to be in a different spot in the sky each day—the orbit of Earth around the Sun and the tilt of the Earth’s axis. When the Earth is traveling along its elliptical, or oval-shaped orbit, and is closer to the Sun (winter in the Northern hemisphere) it speeds up slightly. During the summer in the Northern hemisphere, the Earth is farther away from the Sun and it slows down slightly (see Figure 2). This change in speed causes the Sun to appear either further left or further right in the sky. If Earth traveled in a circle instead of an ellipse, variation in day length would not exist. The tilt of the Earth’s axis causes the Sun to appear higher or lower in the sky. The Sun and the planets follow a path across the sky called the ecliptic. The view of the ecliptic from Earth is tilted due to the Earth’s 23.5 degree tilted axis. The effect of this tilt produces an up and down position of the Sun’s location in the sky. If the Earth’s axis were not tilted, Earth’s seasons would not exist.

The time kept by clocks and watches assumes that there is no tilt to the Earth and that the Earth travels in a circular orbit. However, as mentioned above, this is not true. There is an increasing or decreasing difference between the actual position of the Sun in the sky (or solar time) and clock time. This difference between clock time and solar time is called the equation of time. In the Northern hemisphere, if the Sun’s position is to the east of where clock time indicates it should be, the equation of time is negative. If the Sun is to the West, the equation of time is positive.

The declination angle of the Sun is the angle that the Sun makes with the plane of the ecliptic throughout the year. This angle will vary from 23.5 degrees North Latitude on June 21st (the Summer Solstice) to 23.5 degrees South Latitude on December 21st (the Winter Solstice). On March 21st (the Spring Equinox) and September 21st (the Fall Equinox) the Sun’s declination angle is 0 degrees. In this activity the Sun’s altitude (or the angle of the Sun from the observer’s viewpoint) will be directly measured to gain further insight on the Sun’s declination angle. This will be done using a device called a gnomon. A gnomon (pronounced no mon; Greek for “the one who knows”) is also known as a “shadow stick” and was the earliest form of a sundial. The gnomon was the first time-telling device and was most likely invented by the first person who put a stick in the ground and made marks in the dirt to show where the stick’s shadow was located throughout the day.

Experiment Overview

In this activity, the Sun’s angle in degrees will be measured over the course of a year. The data will be graphed to produce an analemma graph showing the equation of time, the Sun’s altitude and the Sun’s declination angle for any calendar day of the year.

Materials

Safety Precautions

Although this activity is considered nonhazardous, please follow all normal laboratory safety guidelines. Never look directly at the Sun.

Procedure

1. In a room facing the south side of the building, find a location that receives sun at the local noon time throughout the year. If this is not possible, take readings outdoors or in a hallway.
2. Obtain the gnomon paddle, a 1½" wooden dowel rod, and a 2" pin. Place the wooden dowel into the hole of the gnomon paddle. The top of the pin should be exactly 3" above the gnomon paddle (see Figure 3).
   {12787_Procedure_Figure_3}
3. Obtain and place the end of the Sun’s Altitude Sheet directly against the bottom of the shadow stick (see Figure 4). Using transparent tape, tape the Sun’s Altitude Sheet to the gnomon paddle.
   {12787_Procedure_Figure_4}
4. The first data point should be taken on September 20th or 21st.
5. Obtain the sunrise and sunset time for your latitude. These values may be found on the Internet or in a local newspaper. Obtain the local noon time from the instructor or from the Internet. Local noon time is when the sun is at its zenith or the highest point in the sky.
6. Place the gnomon paddle in a North–South orientation on a level location. Use the level on the gnomon paddle to ensure a level location.
7. Using the Sun’s altitude sheet and the location of the shadow from the shadow stick, record the Sun’s altitude angle, in degrees, on the Analemma Data Table.
8. Take your reading. As an example, if your latitude is 40 degrees North, then the Sun’s altitude will be 50 degrees on September 21st.
9. Place this value (in the example below it is 50°) on the Analemma Graph under the Sun’s Altitude heading to the right of the 0 degree Fall/Spring Equinox line (see Figure 5). This 0 line represent the Equator latitude.
   {12787_Procedure_Figure_5}
10. Obtain and refer to the Equation of Time Chart master. Locate the number of minutes the sun is either fast or slow. Place a dot on the 0 degree declination line at the correct time intersection on the Analemma graph. Label this dot with today’s date. For example, on Sept. 21 the Sun’s declination is 0 and the equation of time is +6.40 fast (see Figure 6).
    {12787_Procedure_Figure_6}
11. As readings are taken in the future, it will be seen that the Sun’s altitude will decrease (the Sun will appear lower to the horizon) in the following few months. Beginning with the altitude reading just taken for either September 20th or 21st, subtract and label 2 degrees of altitude on every other graph line in the right-hand side of the graph towards the bottom of the graph until the 24 degree N declination line is reached (see Figure 7).
    {12787_Procedure_Figure_7}
12. When the Sun’s altitude reading of March 21st is taken, the Sun’s altitude will increase upward from the 0 line.
13. Starting with the Sun’s altitude reading for September 20th or 21st, add and label 2 degrees of altitude on every other graph line on the right-hand side of the graph towards the top of the page until the 24 degree S line is reached (see Figure 8).
    {12787_Procedure_Figure_8}
14. Repeat steps 5–9 in the exact location, on the 1st and 15th of every month. If it is cloudy during any day, take your reading on the next available clear day.
15. Connect the dots after each reading is taken. When completed, the graph will resemble a large figure 8.
16. The materials in this activity may be saved for future use.

Student Worksheet PDF

12787_Student1.pdf