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Local Horizon and Sundials
Rosa M. Ros
International Astronomical Union, Technical University of Catalonia (Barcelona, Spain)
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Summary
The study of the horizon is crucial to facilitate the stu- dents’ first observations in an educational center. A simple model that may be made in each center allows us to make the study and the comprehension of the first astronomical rudiments easier. The model is also presented as a simple model of an equatorial clock and from it, we can make other models (horizontal and vertical).
Goals
t Understand the diurnal and annual movement of the Sun. t Understand the celestial vault movement. t Understand the construction of an elemental Sun watch.
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The Earth rotates and revolves
As it is well known, Earth rotates around its axis, which results in day and night. The rotation axis is what ancient astronomers called the axis of the Earth as it seemed that the sky moved around this axis (the daytime sky and the night sky). But Earth revolves in an ellipse, with the Sun in one of its focus. As first ap- proximation, we can suppose that it is a circular mo- tion (as the ellipse’s eccentricity is almost zero, i.e. the orbit is almost a circle).
Earth needs a year to take a full orbit around the Sun, but it does so in a plane, the ecliptic plane, which is not perpendicular to the rotational terrestrial axis; it is
Fig. 1: Scheme of Earth’s revolution. The angle bet- ween the terrestrial equator and the ecliptic plane is 23.5º. The angle between the rotational terrestrial axis and the axis perpendicular to the ecliptic plane is also 23.5º.
inclined. Specifically, the angle between the rotational terrestrial axis and the axis perpendicular to the eclip- tic is 23.5º. Similarly, the angle between the terrestrial equator plane and the ecliptic plane is 23.5º (figure 1). This inclination causes the seasons. To visualize this phenomenon we are going to build a little model (fig- ure 2).
We illustrate this effect with four spheres and a light bulb, representing the Sun, to be placed in the center. It is good to draw the terrestrial surface to distinguish the equator and the poles. Then, we give some values of distances relative to the sphere’s size that represents the Earth models. In our case, we use 8 cm diameter models. We will get a little square tablecloth or paper that is about 25 cm across the diagonal. We situate the four spheres in a cross shape (each one in front of the other, figure 2) elevated using 4 sticks of 3, 15, 25 and 15 cm of height respectively. The values are calculated so that the inclination of the plane of the equator with
Fig. 2a, 2b and 2c: Distribution of the four spheres representing Earth and the light bulb representing the Sun, in the middle. It is necessary to distribute the relative positions so that the angle of the line from the center of the Sun to the center of the Earth is 23º with respect the ground that represents the equatorial plane.
Fig. 3: Model of the revolution motion that explains seasons. When the Earth is at position A it is sum- mer in the northern hemisphere and winter in the southern hemisphere. When the Earth is at position C it is winter in the northern hemisphere and summer in the southern hemisphere. And when the Earth is at positions B and D hemispheres are equally illumi- nated and equinoxes take place. Then, daytime and nighttime are equal.
Fig. 4a. At the po- sition A it is sum- mer in the northern hemisphere and the Sun is 23.5º above equator. However, in the southern hemisphere it is winter.
Fig. 4b. At the posi- tion C it is winter in the northern hemi- sphere and the Sun is 23.5 below the equator. However, in the southern hemi- sphere it is summer.
respect the ecliptic plane is about 23º.
We will situate the model in a dark room and turn on the light bulb (it could be a candle, but always be aware that the relative heights are important). It is obvi- ous that the sphere at position A receives more light in the northern hemisphere than the one at the position C (figure 3), while the illuminated area of the south- ern hemisphere is greater in C than in A. At positions B and D, both hemispheres are equally illuminated; these correspond to spring and autumnal equinoxes. At the times when there is more illuminated area we say that it is summer and when there is less, it is winter. We deduce that when the Earth is at position A, it is summer in the northern hemisphere and winter in the southern hemisphere.
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When the Earth is at position C, it is winter in the northern hemisphere and summer in the southern hemisphere.
This model offers many opportunities for study be- cause if we imagine that a person lives in one of the hemispheres, we will see that he/she sees the Sun in different heights depending on the season. We imag- ine, for example, that we have a person in the northern hemisphere when we are at position A, this person sees the Sun above the equatorial plane 23.5º (figure 4a). However, if he/she is in the northern hemisphere but in the position C, he/she sees the Sun below the equa- tor at -23.5º (figure 4b). When he/she is at positions B and D, he/she sees it exactly on the Equator, i.e. 0º
above the equator. It is not easy to imagine how this model would work, so we are going to build a more realistic model, where the observer is tied to Earth and has no option to see the scheme from the exterior of the terrestrial orbit. We will build a model relative to the local horizon of the observer, AN OBSERVA- TIONAL MODEL.
Observation
Teachers from different science fields (mechanics, elec- tricity, chemistry, biology, etc.) tend to say that it is not possible to work correctly in a secondary science center without a laboratory. In this sense, astronomy teachers tend to be happy because they always have an astro- nomical laboratory. All institutes and schools have a place where students play: the outdoor playground or yard. But these are not only playtime places, they are also astronomical laboratories: a place that offers the possibility to carry out practical astronomical activi- ties. If we have a laboratory in every school or institute, it seems opportune to use it!
Fig. 5: Classical representation of the celestial sphere.
Fig. 10: Model with horizon ring and polar axis.
Fig. 11: Model with the local meridian.
Fig. 12: Sunset point the day of the spring or autumn equinox.
north and the North Pole results easily. Later, we can draw the North-South straight line in the model and also in the court or balcony ground where we took the pictures (using the normal process to determinate the north-south straight line). This is very important because every time we use this model, we will have to orient it, and it is very useful to have this real north- south straight line to facilitate the work. (We can ver- ify this direction with a compass).
The next step consists of locating the meridian of the place. The local meridian is very easy to define, but it is not a simple concept to assimilate for the students (maybe because everyone has his own meridian). We can fix a wire that passes for the cardinal points north and south and the rotation axis of Earth (figure 11). This wire is the meridian visualization of the location of the model, but allows us to imagine the local me- ridian line in the sky. Now it is very easy to imagine because it begins in the same places that student sees in the model. The local meridian begins in the same building as it does in the photo but on the real hori- zon. When the meridian passes above his head, it will end in the same building that we see, thanks to the wire in the horizon of the photos.
The process to introduce the equator is more compli- cated. One possibility consists of the east-west line. This solution is very simple, but it does not reach any- thing from the pedagogic point of view. For educa- tional purposes, it is more convenient to use photog-
raphy again. We can situate the camera on the tripod again in the same position that it was in when we took the first photos of the horizon. (This is why we painted the corresponding marks on the ground, so we could situate the tripod in the same place again). With the camera on the tripod, we take a photo of the sunrise and the sunset on the first day of spring and autumn. In this case, we will have two photos of the precise position of east and west cardinal points respectively, with respect to the horizon in the photos and obvi- ously above the real horizon.
We simulate the equator with a wire perpendicular to the terrestrial rotation axis; it is fastened at the east and west cardinal points (in the horizontal plane that is perpendicular to the north-south line). However, it is not easy to fix this wire to the wire that symbolizes the rotation axis because it is inclined, and obviously it is inclined to the equator also. This leaves the ques- tion as to what inclination to use. We will take four or five pictures of the sunrise on the first day of spring or summer. Photographing the sun is dangerous when it is quite high in the sky, but it is safe during sunrise or sunset when the Earth’s atmosphere acts like a fil- ter. We will use all the photographs and use the ap- propriate software on put them together (using some reference to the horizon), and we can distinguish the inclination of the sun itself on the horizon. This pic- ture will serve to introduce the proper slope on the wire representing the equator in the model (figure 13). Using the two photographs of the cardinal points East and West, it is possible to know the inclination of the traces of the stars in equator, and therefore it is possible to locate the wire that symbolizes equator smoothly. We now know the fixed points and also the inclina- tion, so the wire can be fastened on the frame and also hold the local meridian (figure 13).
If we consider the Sun as a normal star (the Sun is the most important star for us because it is the nearest,
Fig. 13: Trace of the sunrise.
Fig. 14: Traces of the stars in the east.
but its behavior is not very different from other stars), we can obtain the inclined motion of stars when they rise or set with respect to the horizon. To do this we only have to take two pictures of this instant near the cardinal point east and west (figure 14).
It may be impossible to take the pictures mentioned in the previous paragraph from the city where the school is built. We have to go to the countryside, in a place that is not affected by light pollution, and take pictures with a single-lens reflex camera on a tripod with a cable release. About 10 minutes of exposure is enough. It is very important to place the camera parallel to horizon (we can use a level to do this operation).
Take this opportunity to get a small portfolio of pho- tographs. For example, you can take one of the pole area giving a 15 minute exposure, another one of the area above it along the local meridian, another one fol- lowing the same meridian and so forth, until you get to the picture that is on the horizon. The idea is to photograph all the local meridian from north to south, passing over our heads. Obviously, the local meridian of the place where we have decided to take pictures is not the same as that of the school, but students can easily understand this small difference.
When we have all the pictures, we can build a me- ridian strip with them all. With this strip, students can better understand the movement of the celestial
sphere around Earth’s axis of rotation. Interestingly, with the same exposure time, the trajectories drawn by stars change their length. It is at a minimum around the pole and maximum at the equator. It also changes shape. At the equator, the trajectory draws a straight line. In the area near the pole, lines are concave curves above the equator and are convex below. If we make paper copies of the pictures large enough, we can put the strip over the head of the students, allowing them to visualize and understand the movement better.
Using the two photographs of east and west cardinal points, it is possible to know the inclination of the traces of stars at the equator, and therefore it is possible to locate the wire that symbolizes the equator without problems. We know the points where we have to fix it and also the inclination, so the wire can be attached to the wood and to the local meridian (figure 8).
It is clearly possible to introduce the strip of pictures of the local meridian on the model. It is sufficient to make some copies and make a hole in them at the point that indicates the pole, in order to introduce the axis of rotation. Note that the wire of the equator cor- responds to the straight-line traces that are on the tape (figure 15).
With this model, we can offer the students the two possibilities of viewing the celestial sphere from the in- side and from the outside.
If we again take two pictures of the first day of win- ter and summer when the Sun rises and sets, students will be able to see that the locations are very different in their city. The difference between them is amazing. You can also set the parallels of Cancer and Capricorn with the pictures that give the slope of the equator, since the parallels follow this same inclination. With a simple conveyor, it is possible to verify that the internal angle between the Tropic of Cancer and the equator is about 23º, and this is also the angle formed between the equator and the Tropic of Capricorn (figures 16 and 17).
Fig. 15: The local meridian pictures.
Fig. 20: Equato- rial clock in used in northern hemi- sphere.
Fig. 21: Equatorial clock used in south- ern hemisphere.
Fig. 22a, 22b, 22c and 22d: Some images of the clocks.
primavera - verano otoño - invierno
primavera - verano otoño - invierno
evident in the model that the number of hours of solar insolation is greater. The natural consequence is that it is warmer in summer than in winter (figure 19).
We will take this opportunity to mention some ele- ments that must be known to construct a sundial.
The equatorial clock is very easy to create. Just put the stylus in the direction of Earth’s rotation axis, i.e., in the north-south direction (a compass can help us do so), and with a height above the plane of the horizon equal to the latitude of the site (figure 20 and 21). The stylus of any clock always will be placed in the same way.
The equatorial clock hour lines are drawn at 15 degrees (figure 23), since the Sun gives a 360 degree turn in 24 hours. If we divide 360 by 24, we get 15 degrees each hour.
The hour lines of a horizontally or vertically oriented clock are obtained by projecting the equatorial lines and simply considering the latitude of the place (fig- ures 22a, 22b, 22c and 22d).
Solar time and clock time of wristwatches
Sundials give solar time, which is not the same as that on the watches that we all use on our wrist. We must consider several adjustments:
Longitude adjustment Earth is divided into 24 time zones from the prime meridian or Greenwich meridian. To make the longi- tude adjustment it is necessary to know the local lon- gitude and the longitude of the “standard” meridian in your area. A “+” sign is added to the east and signed “-” to the west. We must express the lengths in hours, minutes and seconds (1 degree = 4 minutes).
Summer/winter adjustment Almost all countries have a summer (“daylight sav- ings”) and winter times. An hour is usually added in the summer. The time change in summer/winter is a decision of the country’s government.
Time equation adjustment Earth revolves around the Sun according to Kepler’s
law of areas for an eclipse, i.e., it is not a constant mo- tion, which creates a serious problem for mechanical watches. Mechanical clocks define the average time as the average over a full year of time. The Equation of Time is the difference between “Real Solar Time” and “Average Time”. This equation is tabulated on Table
Fig. 23a and Fig. 23b: Cut of the equatorial clock.
Fig. 25: The large-scale model in the Science Park of Granada.
they previously would not have. For example, now it is very clear that the Sun does not rise and set perpen- dicular to the horizon except at the equator.
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Bibliography
Ros, R.M., “De l’intérieur et de l’extérieur”, Les Cah- iers Clairaut, 95, p.1-5, Orsay, 2001. Ros, R.M., “Laboratorio de Astronomía”, Tribuna de Astronomía, 154, p.18-29, 1998. Ros, R.M., “Sunrise and sunset positions change every day”, Proceedings of 6th EAAE International Summer School, 177, 188, Barcelona, 2002. Ros, R.M., Capell, A., Colom, J., El planisferio y 40 actividades más , Antares, Barcelona, 2005. Ros, R.M., Lanciano, N., “El horizonte en la As- tronomía, Astronomía Astrofotografía y Astronáutica”, 76, p.12-20,1995.
