Solving problems with a solar motion demonstrator

Werner Warland

"EAAE Summerschools" Working Group

Schlossgymnasium Benrath, Düsseldorf, (Germany)

Abstract

The sun is the source of essentially all energy that drives our climate system on the earth. The earth only gets a very small fraction of the sun's total output, because the two bodies are far apart. The distance varies throughout the year, and the sun's output is not constant. The actual amount of radiation received at any part of the surface at any given time is controlled by many factors. In this lesson we will consider some of the factors related to the position of the sun with respect to a location on the earth. We will look at the effects of the day of the year, the time of the day, and the latitude.

To study Positional Astronomy each participant will construct his own celestial sphere with the zenith above his head and a horizon N-E-S-W. From a cardboard, glue and brass fastener each one is building his own small, hand-held device, that is to demonstrate the apparent mo-tion of the sun across the sky at any time of the year for any observer at any latitude in the northern hemisphere. The participants are investigating a collection of problems with the instrument's use.

They can research questions like:

How high above the horizon is the Sun at a certain date?

What is the apparent movement of the sun at the equator or at a special location?

How many hours will the sun be above the horizon on any given day?

What happens on a solstice? On an equinox?

Observing, experimenting, predicting and visualising skills are developed. The skills have real-world application. The device is a good preparatory tool for visiting a planetarium, for introducing positional Astronomy, and for working with an armillarsphere in a classroom.

1. Assembling the Solar motion device

With scissors cut out the frame along its outline. Fold the frame along the creased lines, so that the month piece swings all the way around. Fold the flap marked „glue" and the quarter-circle below it away from you so that a right angle is formed. The backside of the blank quarter circle hits the backside of the frame.

Cut a small slot in the horizon disk at north position. Apply glue to the marked portion of the frame. Press the north east quadrant of the disk against the glued proportion. The correct alignment of the frame and the disk is essential to the working of the device.

The head of the brass fastener represents the sun. Bend the fastener tabs over the outside edge of the month piece. The fastener must be able to slide up and down along the month piece.

The Solar Motion demonstrator was designed by Professor Snider of Oberlin College, USA. You may reproduce it for your own classroom or planetarium use (but not for commercial purposes).

2. Using the Solar Motion Demonstrator (SMD)

- The edge of the horizon disk represents the visible horizon. You can imagine a tiny person standing at the black dot in the centre of the disk.

- The head of the brass paper fastener represents the sun.

- The latitude part of the frame is used to adjust the horizon disk and to set the observer at any latitude from the Equator (0°) to the North Pole (90°).

- The twisted month arm of the frame has two functions: Setting the sun marker at the desired month adjusts for the time of the year. Swinging it from the east to the west moves the sun in its apparent daily path over the earth.

3. Activities

Dates for use

3.1. Basic understanding

I. Questions using your home location

Justify your answer using SMD.

1 - How high is your Local Noon?
Imagine you are standing at the black dot of the horizon disk. A clear horizon around you. With the other hand you pivot the month-semicircle. The paper fastener describes the path of the sun. The angular height of the sun is the angle between your line of sight to a point on the horizon directly beneath the sun and your line of sight to the sun. The sun reaches its greatest height at a time halfway between the sunrise and sunset. By changing the fastener to different months you can get a sense of how large this maximum angular height is for the various times of the year.

2 - Is Daytime as long as Nighttime?
If you pivot the fastener over its entire range, this corresponds nearly to one rotation of the Earth, which takes 24 hours. You can determine the relative lengths of day and night by comparing the part of motion above (daytime) and below the horizon.

3 - When are day and night equally long? Are they depending on your latitude?
There are two days called "vernal equinox" (about March 21) and "autumnal equinox" (about September 22).

4 - When will the sun be at the lowest or highest altitude in the sky?
These days are called the "Summer Solstice" and the "Winter Solstice". On these days the fastener stops and reverses its direction of motion along the month piece. The word "Solstice" means "Sun stands still".

II. Questions for points at the Northern Hemisphere

1 - Why does the Earth have seasons?
Two factors are responsible for seasons ( referring to the horizontal system ) : the length of the day, and the angle sun's rays strike the ground.

2 - Are there places on Earth where Sun doesn't set?
North of the "Arctic Circle" (about 66,5° latitude) the sun will not set at least one day of he year.

3 - What path does the Sun take at the Equator?
Latitude 0°: Vary the time of the year and see how the path of the sun across the sky changes. Notice that the setting sun moves in the same way.

4 - What is the motion of the sun for an observer at the North Pole during a year?

5 - When and where does the sun pass the Zenith?
The point straight ahead on the celestial sphere for any observer is called the Zenith and is always 90° from the horizon. The arc that goes through the north point, Zenith, and the south point of the horizon is called Meridian.
Explore the range of latitude and of times of year for which the sun passes through the zenith. For an observer north of the "Tropic of Cancer" ( at about 23,5° north latitude ) the sun will pass through the zenith. For lower latitudes, it will pass through the zenith at two days of the year only. Which are these days?
The Solar Zenith Angle (Z), which is the angle between the sun's rays and the zenith direction, is the numbers of degrees, that the sun is away from being directly overhead. The complementary angle (90-Z) is the Solar Altitude (or elevation). This is the number of degrees that the sun is above (or below) the local horizon. Solar Noon is that time of day when the sun has reached its highest position in the sky at your location. It does not usually coincide with the noon at your wristwatch.

6 - How to make a compass without a magnet needle.
Set SMD to your latitude and time of the year. Go outside in the sunshine. Hold the horizontal disk of SMD horizontal. Pivot the month part and turn the horizon so that the "N-S" line points in various directions. Hold the device in a way that the shadow of the month part be a thin line as possible, while at the same time the shadow of the paper fastener (=Sun) falls exactly in the middle of the horizon disk, which will show you now the correct geographic directions.

III. More detailed Questions

1 - Determine the azimuth for sun raising and sun setting depending on the seasons at your home location.

2 - The North Cap is a 307 m high rock above the sea. Determine the dates of seeing the midnight sun completely.

3 - Astronomers define the Civil Twilight as a time which lasts from the sunset to the time, at which the middle of the sun reaches the angle of the depth of 6°, when the sky is clear.

1 - At which time of the year the twilight is shortest, when it lasts longest?

2 - Discuss the meaning, in the Tropics twilight time lasts much shorter than in Europe, the day changes from one minute to the next into the night.

3 - Interesting is also the twilight in polar regions. How long and how short lasts the twilight at the Arctic Circle or the North Pole

3.2. Increasing understanding

(justify your answer using the SMD)

1. If your room's window (in the Northern Hemisphere) faces North, will you be able to see the sun through it? If true, in which moment of the day and what time of the year it is happening?
2. A scientist working in a base in Antarctica, at latitude 90° South, and questioning himself: which will be the highest altitude of the sun, and on which date? What will happen to the fur of the polar bears that day?
3. If you start at Santa Cruz (Galapagos Island 90°W, 0°) in eastern Direction and you fly along the geographic equator in a plane at noon on Christmas Day, on which window the sunlight gets into the plane?
4. In a paragraph of a certain novel reads „Whilst Venus shone high in the sky, the square clock sounded midnight". Can this be true?
5. Why are there so many mosquitoes in Finland in summer? Remember that mosquitoes need continuously warm conditions. Verify your answer by using SMD.

4. Basics of Positional Astronomy

The apparent motion of the sun results from the rotation of the earth every 24 hours around its axis. Even though it is known that the ancient model of a stationary Earth is incorrect, we will still use it because it is a convenient way to predict the motion of the Sun relative to a location on the Earth. It coincides with the every-day experience of the children.

An old German poem says:

The sun rises in the East.

It reaches its climax in the South.

It sets in the West.

But it will never be seen in the North.

This statement is only correct at the equinoxes. On both days at any point of the earth the sun rises exactly at 6 a.m. in the East and sets at 6 p.m. in the West. At any other day the rising and setting points are shifted to the North and South depending on the latitude.

If you specify the location of the sun in the Altitude - Azimuth System, the observer is located at the centre of his "celestial sphere" with zenith Z above his head and the horizon N-E-S-W.

Any celestial body can be identified by two co-ordinates: Altitude h and Azimuth a (horizontal co-ordinate). The Altitude h is the angular distance above the horizon (0 < h <90°), and the Azimuth a the angular distance, measured along the horizon, westwards from the South Point S (in Astronomy) or eastwards from the North Point N in Nautical Science (0< a < 360°). Observers at different locations looking at the sun at the same time will see it at a different altitude-azimuth position. The daily movement of an object - resulting from the rotation of the Earth on its axis - starts at (1) when it rises. At (2) it passes across the observer's meridian NZS, reaching its maximum altitude above the horizon (transit, culmination), and it sets at (3).

Select the Java applet "Apparent motion of a star" to show an animation of the celestial sphere changing with latitude.

Download it!

The second way of specifying star positions -preferred by astronomers- is the Equatorial Co-ordinate System. This system is very similar to the Longitude-Latitude System used to specify positions on the Earth's surface. This system is fixed with respect to the stars. A star's position does not depend on the observer's location or time. Fig.4 shows, how in the Equatorial System Right Ascension (RA) and Declination δ are defined.

Sun's Position:

On its daily path in the sky the sun reaches its maximum height exactly in the south.
The meridian altitude of the sun = 90 - (observer's latitude) + δ      (Eq. 1)
The sun's declination d ranges between -23,5° and +23,5°. It changes fairly slow around the Solstices and fairly rapidly near Equinox.

At the vernal equinox sun rises at 90° azimuth and sets at 270° azimuth (RA = 0h; δ =0°).

At June solstice sun rises at less than 90°azimut and sets at greater than 270° azimuth (RA =6h, δ = +23.5°).

At autumnal equinox sun rises at 90° azimuth and sets at 270° azimuth (RA = 12h; δ = 0)

At December solstice sun rises at greater than 90° azimuth and sets at less than 270° azimuth (RA = 18h; δ = -23,5°).

Celestial reference marks:

The celestial equator always intercepts horizon at exactly East and exactly West points. The sun moves parallel to the celestial equator.

The altitude of celestial equator on meridian makes with horizon an angle of 90° observer's latitude.

The sun's declination d equals the distance to the celestial equator.

Chapter 4 may help to understand the construction of the SMD. But it is not necessary for the use of the device.

5. Some Activities concerning Positional Astronomy

For the following activities alter the design of SMD. The "month arm" has to be covered with a blank strip of paper. If you put an set square at the horizon, you can determine the solar altitude by measuring the angle.

1 - Comparing the results of modelling and of exact calculation
At which days sun crosses the meridian at its highest and its lowest point in Helsinki, Bad Honnef and Tavira? How long last those days at the different locations?
By twisting the month arm determine the meridian points, find out the angles and compare the results with the calculated meridian altitudes of Eq. 1.

2 - Observing Midsummer Days North of the Arctic Circle
Using observations of the solar altitude during a midsummer day you get the maximum and minimum elevation at Your location. If h_sun,max is the Meridian Passage of the sun in the South and h_sun,min means the Meridian Passage in the North you can determine the declination of the sun:
h_equator,max + δ = h_sun,max
h_equator,min + δ = h_sun,min
h_equator,max = - h_equator,min
h_sun,max + h_sun,min= 2δ
δ = (h_max + h_min) / 2      (Eq. 2)
Determine observer's latitude by marking the known values on the blank month's arm of SMD.