**Felisbela Martins ^{1}, Rosa M. Ros^{2}**

**"EAAE Summerschools" Working Group**

^{1}University of Porto (Portugal)

^{2}Technical University of Catalonia (Spain)

# Abstract

For most of history, man's days and years were governed by the sun and stars. Even after clocks were invented, they were set by the sun. In our days we know about planets, stars and galaxies but we have lost touch with the logic and rhythm of the sunrise, sunset, summer and winter. It is instructive to learn how a sundial works and watch its shadow advance through the day and vary with the seasons. A sundial consists of the dial plate marked out with hour lines, and a gnomon, the raised projection that casts the shadow. The inclined edge of the gnomon, called the style, produces the work edge of the shadow that it is used to tell the time. It is oriented parallel to the earth's axis, pointing toward the point in the sky around which the (imaginary) celestial sphere rotates once every 24 hours, which is very close to the location of Polaris.

There are many different types of sundials. Virtually anything casting a shadow can be made into a sundial; the trick is to calculate the proper placement of the time marks. To be accurate, a sundial must be especially designed for the spot it is to be used in and must also be pointed in the right position.

The main aim of this workshop is to built a simple sundial. The pedagogical aims of this construction is allow the participants and their students to understand the set going of the sundials, how determinate the hours, how to find the right place and the right orientation for a sundial and identify different types of sundials.

It will be very interesting that each participant built a sundial for the latitude of is own school and latter on put in practice the workshop with their own students.

# Introduction

This task is intended to promote the creation of a European school network, which, inside the classroom or in extracurricular activities, will motivate students into building sundials. As schools start taking part in the project, the sundials will be displayed on the web page of European Association for Astronomy Education.

Assuming that not all teachers are familiar with these concepts, very simple activities are proposed, which support teachers of various age levels, in the teaching-learning process of basic concepts such as: the apparent, daily movement of the sun; cardinal points; height of Polar Star, midday sun and mean time.

# Sundials

From very early on Man understood that any object under the sunlight produces a shadow, and that this can indicate how much longer there will be daylight for. That is how Man invented the sundial. This is based on the sun's apparent movement around the celestial vault and in the consequent moving of the shadow projected on a plain or curved surface, of a body illuminated by this star.

This is a fixed or portable tool aimed at determining the time of the day through the movement of the shadow produced by a straw or any other object under the sunlight.

# Activity 1: The position of the Sun in the sky

This is a starters activity, aimed at younger students or those with very little knowledge of the subject. It is important to remind the students that the sun seems to move slowly in the sky during the day. The main problem in this kind of project is the time required for the tasks and choosing days in which the sun is visible.

One should choose somewhere on the Southern side of the school, where the sun can be seen without impediment during a period of time preferably between 9h and 15h. (It is important to make the students aware of the dangers involved in staring directly at the sun). It is also important that students make a note of facts and observations.

They should note down the direction, in which the sun can be seen, for example "It is over that tree" or "It is over that building". It is important that all the students perform their observation in the same place. They should note down their observation of the sun's position. The complete notes are then used to compare the different positions of the Sun and deduce why it "moved" slowly.

# Activity 2: Poster

After recognising that the sun "moved", students can make a poster individually, but mostly in groups. The poster should represent the landscape on the southern part of the school, signalling the positions of the Sun. Each time an observation is made, "THE SUN" is shown in the adequate position. At least three, time-distant observations should be made during the day, for example, at 9h, 12h and 15h. If students stay at school until later, they can make a final observation at sunset.

# Activity 3: The shadows

The shadow, which is most familiar to us, is our own. Students can start by exploring the behaviour of the sun-projected shadows, just by standing up outside. Many simple objects can be used as sundials.

Place a straw on a big white piece of paper, somewhere outside school where there is sunlight all day long. Do not touch it again. Visit the place five or six different times during the day. Draw the straw's shadow on the piece of paper on each visit, captioning it with the time and being careful not to move the straw or the piece of paper. The complete observations produce a poster (which can be used as clock in the following days) showing that the length of the shadow and its direction vary as the day goes by. At 12 noon, the shadow will be shorter, and at the beginning or at the end of the day the shadow will be longer. Draw a line, which follows the various registered points. Place yourself in the central mark again and turn to the point over the line that is closer to you. At this moment you have just turned to the North Pole of the Earth. This can be verified with a compass. If you wish, draw the cardinal points on the piece of paper or on the floor. The Sun is higher up in the sky at midday. At sunrise its shadow is longer. You can therefore build a sundial, which allows you to know the time of the day.

Questions such as "What produced the straw's shadow?" or "Why did the shade move?" or "What time was the shade shorter?" can be developed with the students.

# Activity 4: Solar midday and mean time

In reality, sundials do not tell the time as mechanical clocks do. The latter divide the day into exactly 24 hours, each hour in 60 minutes and each minute in 60 seconds. It is a division calculated by Man, and the period represented by the 24 hours of a mechanical watch corresponds to an average solar day. Sundials, although they mark the time approximately as mechanical clocks, rarely coincide with them, given that they show the time according to the real solar midday- true time or true solar time. Is the time, which the Earth takes to complete one complete turn around its axis, using the sun as a reference point. What happens is that whilst the Earth turns around its axis, it goes through a part of its revolution orbit around the Sun. The Earth, turning around its axis, moves across Space, i.e. it completes two movements: rotation and revolution.

Returning to the example provided by the previous activity, it is solar midday at school when the straw's shadow (gnomon) is shorter. In that instant, the shadow is on the place's meridian. At the moment in which the shadow is shorter, is it solar midday. In our regions of the North Hemisphere, at solar midday, the Sun shows us South. It is therefore a way of positioning south and, consequently, the remaining cardinal points. The activity can be developed in the following days, enhancing the notions of solar day (interval between two consecutive days. This is divided in 24 parts called the hours. The middle of the night - midnight- corresponds to the instant 24 hours or 0 hours, and the middle of the day, or midday, corresponds to the instant 12 hours.

# Activity 5: Building an equatorial sundial

By building an equatorial Sundial, one can realize the regularity of the sun's apparent movement through the day. It applies to any part of the globe. The equatorial sundial is generally done on a disc with a perpendicular hand, which spreads across from side to side. It should be turned to the pole. The hours are marked in 15 degrees intervals.

The Earth turns around its imaginary axis in a 24-hour period. When performing this movement, the Earth rotates 360º. Therefore, during one hour it rotates 15º. Let's imagine we are in the North Pole. We place a disc on the floor, divided in 15º sections; it is possible to know the time by using the shadow of a vertical straw, placed in the centre of the disc. As soon as the Sun passes the meridian of place one knows that the shadow will reach the following sector in one hour and so on.

Let's imagine we move the disc with its straw and respective 15-degree intervals in another point of the globe with certain latitude. We must be careful not to change its alignment with respect to the Earth's meridian and axis. We then obtain an inclined disc, from which a straw comes out, directed at the poles. In this case, our sundial's disc in the equatorial plane and the straw is parallel with the world's axis, forming and angle with the place's horizon.

The top is lit from the spring equinox until autumn equinox, while the lower part is lit from the autumn equinox until the spring equinox.

**To build an equatorial sundial, the following steps should be taken:**

1. A wooden board, or one of any other material.

2. Draw a circle with 20 a 50 cm diameter.

3. Saw or have a circle sawn over the circumference line.

4. Draw a straight line over the centre of circle (diameter) and mark a point precisely at the centre of the diameter.

5. Make the central point in the protractor and the black line, which goes from 0º to 180º, coincide with the circle's diameter line and the point marked on it.

6. Holding the protractor firmly, signal, with a pencil, the degrees every 15º (from 0º up to 180º).

7. Lift the protractor and, using a ruler and a pencil, link the point marked at the centre of the circle's diameter with each mark you signalled and keep going until the edge of the circle.

8. You can also mark rays with 1,25º angles for 5 minutes.

9. These rays should be drawn on the two faces of the disc so that they are all in the same direction and over each other.

10. With a hole punch, make a hole in the point marked at centre of the diameter of the circle and as wide as the straw (metal/Wooden, straw) which will be introduced in that hole.

11. Draw the numbers of the hours on the face and decorate the sun dial as you wish (if it is wooden, varnish it to waterproof the final surface).

12. Pass the straw through the hole at the centre of the sun dial with an angle inclination which is the same as the place's latitude, in relation to the ground (look at the image and use the protractor to incline the sun dial, using the ground as the base. Having done this, try to stick the straw in this angle on the sundial (with glue or a nail).

**Positioning the sundial:**

1. On a horizontal plane, with a compass, mark East-West and North-South directions.

2. Place a wooden plane, that will be used as the plate along the E -W axis. Place the straw along the N-S axis and facing north.

3. Move the straw so that with the N-S meridian it creates an angle at the determined height of the Polar Star

4. This is a removable part. All that is necessary is to establish the angle between the foot of the straw and the meridian, i.e. polar height.

# Activity 6: Building rectangular or square Equatorial Sundial

It is done in the same way as the previous one; the only difference is that of the sundials format. One can draw the sundial directly on the wooden surface, which in this case is a rectangle or a square.

# Activity 7: Building an interior Sundial

This activity is aimed at students and consists of building a Sundial to be placed inside a house. It also serves the purpose of consolidating and/or assessing students' knowledge.

**Material:**

Cardboard for the base, and card

12 cm long wooden straw, or a straw

Pencil, eraser and ruler

Glue and scissors

Protractor

Material of your choice for the decoration

Self-adhesive transparent paper / or laminated plastic to waterproof and protect the sundial

Given scheme.

**Assembling:**

1. Copy the above scheme in the desired size in card and cut out the sundial.

2. Make a little hole on the black dot for the straw.

3. Use your imagination to decorate your Sun dial.

4. Cover the sun dial with self adhesive transparent paper / or laminated plastic.

5. On the card, draw two similar rectangle triangles, proportionally to measurements of your sundial - one of the angles has to be the same as the measurement of the latitude of the place where you are. Cut the triangle and fold along the dotted lines. They will be placed in the lower part of the sundial.

6. Lean the straw so that its lower part fits in with the paper, which has been lifted.

7. Place the Sundial in a well-lit place, by the window or a balcony. With a compass, orientate the Sundial according to N/S points- the end of the straw should point to the North Pole.

# Activity 8: A simplified analemmatic sundial for your school

Why not build a sundial in the courtyard of your school? This is a nice activity, which can involve a big group of people from the school! In general students are keen to decorate the sundial they have seen in the making. The next question is what kind of sundial? The gnomon is the most difficult part of a sundial to be conserved in a public place or in a school court due to its delicate nature. Weather problems, animals and children can change its position effecting the readings of the sundial. In these kind of places the best solution is to build an analemmatic sundial. In this case, the plane of the sundial is horizontal, the soil, and the gnomon is the observer, that is to say the person who is interested in reading the time. Of course, then the gnomon is not oriented according to the rotation axis of the Earth. In these kinds of sundials, the gnomon is perpendicular to the horizontal plane. It is necessary to change the gnomon position every day, but this is not a problem for the reader. He/she can change his/her position if we indicate it clearly. Obviously, the best solution for building a sundial in your school courtyard is to make an analemmatic sundial. Below we will explain how to draw one of them on a piece of paper, and it would only be necessary to modify the scale to paint it on the soil at your school. The process is:

1. Draw a horizontal straight line and draw the latitude of the place f using a protractor. On the horizontal line draw the segment a, according to the dimensions that you decide for the sundial (in total the maximum size will be 2a). W and O are the two extreme points (figure 1). W is the side of the angle vertex. The inclined segment is c.

2. Draw a right-angled triangle with hypotenuse a. Place a ruler along the inclined line c. Take a set square and position it´s cathetha alongside the ruler. Move the set square until the other cathetha arrives at the point O and draw a line to complete the triangle. Name WOB this triangle (figure 1).

3. Draw a circumference of radius a with the centre O, which passes through point W.

4. Draw two perpendicular diameters through the centre O. One of them is the line WE and the other line NS. When you paint the sundial on the soil it is necessary to orientate both lines using the compass or, if it is possible, using the meridian line.

5. Name b the segment OB and c the segment WB of the right-angled triangle. Both values will be useful for drawing the ellipse of the hours of the sundial.

6. Draw from the centre O the distance b to determine the minor semiaxis of the ellipse on line NS (figure 1).

7. Draw from the point O the distance c to both sides of the diameter WE to determine the ellipse's focus F and F´ (figure 1).

8. Draw the ellipse using the focus F and F´ where you fix the two ends of a piece of string of longitude 2ª. Using the method of gardeners', draw the ellipse. You will use the same process when you translate the sundial from the paper to the soil (figure 2).

9. Divide the circumference into 24 parts each one at 15º (figure 1). Mark these points on the circumference.

10. Draw a set of lines, parallel to the diameter NS, from these points to connect with the ellipse. These intersection points will be the points to place the hours. 12 o'clock is positioned at the North and the other hours are placed one by one until 6am is positioned at W and 6pm at E.

11. The position of the gnomon, that to say the position of the person who will use the sundial, is on the line NS. It is well known that this position changes according to the annual zodiacal positions. In particular, on March 21st and September 21st (Aries and Libra points) the gnomon will be in the NS line intersection with the WE line (point O) (figure 3).

12. Draw an angle of 23.5º centred at the focus F on the line WE in the northern semiplane. The intersection of this line determined by 23.5º with the line NS will mark the extreme position of the gnomon on June 21st (Cancer) (figure 3).

13. Draw an angle of 23.5º centred at the focus F on the line WE in the southern semiplane. The intersection of this line determined by 23.5º with the line NS will mark the extreme position of the gnomon on December 23rd (Capricorn) (figure 3).

14. Draw the intermediate positions. An angle of 20º centred at F on the line WE in the northern semiplane determines the gnomon's position on May 21st (Gemini) and also July 23rd(Leo), in the southern semiplane the same angle determines the gnomon's position on January 20th (Aquarius) and also November 22nd (Sagittarius) (figure 3).

15. Finally, draw an angle of 11º centred at F on the line WE in the northern semiplane determining the gnomon's position on April 20th (Taurus) and also August 23rd (Virgo). Drawn, in the southern semiplane, the same angle determines the gnomon's position on February 18th (Pisces) and also October 23rd (Scorpio)(figure 3).

16. For more details you can mark the points of several days each month. In this case use the solar declination for each day.

17. Translate the completed plan of the sundial to the soil placing the line NS with the median line.

# Activity 9: A simplified solar calendar

It is possible to build a simple calendar using the annual solar path each day at twelve o'clock. In fact, this is a partial study of the previous analemmatic sundial. The method is:

1. Cover a window of the classroom with a piece of thick paper or a thin piece of wood, in which you have made a circular hole for a ray of sun to pass through.

2. Draw a line on the floor of the classroom oriented to the North-South direction. Mark on this line the position of the zodiacal signs. Below you will calculate the distance from the wall for the different signs, depending on the altitude h of the hole from the floor and the latitude of the place f .

3. It is well known the declination of the Sun is zero on March 21st, then the position of this point has been determined by the latitude of the place f and the distance h of the hole to soil (figure 4). The distance from the wall to Aries or Libra (first spring or autumn day) is d =h tan f

4. To determine the position of Cancer (first summer day) is d= h tan(f - 23.5) and the position of Capricorn (first winter day) is d= h tan(f + 23.5) (figure 4). In general the position of all the other zodiacal sign is d= h tan (f - D) . Where D is the Sun declination which appears in table 1, D= -23.5º, -20º, -11º, 0, 11º, 20º, 23.5º.

5. Each day, when the Sun passes over the line, you know that it is midday and you can read which zodiacal sign it is. If you are interested in increasing the precision you can mark on the soil the days of the month at intervals of 10.

# Sundial Time and Clock Time

Sundials tell the local sun-time, which is not the same as clock time. To be able to convert between one type of time and the other we have to consider three different adjustments.

**1. Adjustment for Longitude**

The world has been divided into 14 time zones, each based on the prime meridian at Greenwich. To make the longitude adjustment you need to know your local longitude and the longitude of the standard meridian for your time zone. We add + to the east and - to the west. It is necessary to express the longitudes in hours and minutes (1 degree is 4 time-minutes).

**2. Adjustment for Summer Time**

You must remember to add an hour to the sundial time during the Summer Time period.

**3. Adjustment for the Equation of the Time**

The Earth is not rotating around the Sun with constant speed. It would be most inconvenient if clock time was to vary from day to day and so clock time is averaged over a complete year. This is called the "Mean Time". The Equation of Time is the difference between the "Real Sun Time" and the "Mean Time". This Equation of Time appears tabulated in table 2.

# Time Conversion

*Sundial Time + Total Adjustment = Clock Time*

**Example 1:**

Barcelona Catalonia (Spain) on May 21th.

**Example 2:**

Tulsa Oklahoma (EEUU) on November 16th.

# References

Broman, L., Estalella, R. Ros, R.M., Experimentos de Astronomía, Ed. Alhambra, Madrid, 1988.

Farré, E. Segura, C., 24 rellotges i altres instruments per a la mesura del temps, Instruments Guix, Ed. Grao, Barcelona, 1989.

Jenkins, G., Bear, M., Sundials & Timedials, Tarkins publications, Norfolk, 1987.

Rohr, R.J. Sundials. History, Theory and Practice Dover Pub, INC New York, 1970