
SUNRISE AND SUNSET POSITIONS CHANGE EVERY DAYRosa M. Ros

The inclination of the Earth's rotation axis causes the seasons and the position of sunrise and sunset to change every day. The maximum angular distance between two sunrises or two sunsets is the angle between two solstices. This angle changes with the latitude of the place. It is minimum at the equator (where it is equal to ecliptic obliquity), and after that increases according to the absolute value of the latitude until it causes the midnight Sun in the polar area. The main objective of this workshop is to present this phenomenon through images and also to calculate the inclination of the Earth's rotation axis for each latitude. It is enough to take a set of sunset photos in different latitudes to obtain the ecliptic obliquity with an acceptable error. The secondary idea of this workshop consists of studying the relationship between the solar path, the horizon and the latitude, by means of photography.
The main interest of this workshop is to observe, with a group of photos, that the inclination of the Earth's rotation axis (e=23º 27') causes the duration of the day to change during the year, and as a consequence the sunrise and sunset points change throughout the year. In particular, the first objective was to take a set of photos in different latitudes to show the position changes of sunset points during the year. The photos are the result of cooperation between five different countries (Latvia, Spain, Colombia, Bolivia and Argentina). The same situation appears at sunrise or sunset, so it was decided to work at sunsets because it is always more convenient. 
In an equatorial city (latitude f=0º), the distance between two sunsets can be, at maximum, 2e (between the June and December solstices, (fig 1).
If the place of observation has an intermediate latitude , the distance between both solstice sunsets 2x is greater than 2e (fig 2), because when f increases this distance 2x also increases.
And finally, at the pole the sun path is parallel to the horizon (midnight Sun), and it is not possible to consider the angular distance x, because the Sun does not have sunset points (fig 3).
The purpose of this workshop is to show this phenomenon with a set of photos taken in different places at approximate latitudes 60ºN, 40ºN, 20ºN, 0º, 20ºS, 40ºS and 60ºS. Cities with Northern and Southern latitudes have been included.
The apparent movement of the Sun has peculiarities in each hemisphere (fig 4), but the situation of the sunset points is the same (fig 5). It was not possible to obtain the photos in each city on the first day of each season, because sometimes it was raining or cloudy. This situation was prevalent in places with extreme latitudes (60ºN, 0º or 60ºS), because they have special periods of weather. In these cases the photo was taken on the first day that it was possible. However, in each case, a pair of photographs of the first day of two consecutive seasons was obtained (solstice and equinox or equinox and solstice) to determine the maximum angular distance x, for each latitude. For example, it was very difficult to take the photo of the June solstice in Riga (it was necessary to wait two years, and besides that it was taken on July 4th) because this period is normally very cloudy there. However the photos of the December solstice and March equinox were obtained more or less correctly (December 22nd and March 20th), and this is enough to determine x from the observations (fig 5) .
The photos were used as observations to calculate the angular distance x. After that the obliquity of ecliptic e for each latitude was determinate. From photo 1 it is possible to measure the distance d in cm and it can be converted to x degrees with the annex. Afterwards, it is possible to obtain e using the sine theorem in figure 6,
by introducing the value of the places latitude f, the ecliptic obliquity can be obtained,
In table 1 the results obtained of the angular distance x in each city and the obliquity of ecliptic appear. These results mean a small error justified by the process.
It is possible to observe that, on average, the value obtained is e = 23º.5. It was not possible to take the photo at the instant of the start of the season. The photographic observations introduce an error motivated by the image deformation produced by changing a circular horizon to a flat surface. This phenomenon is especially sensitive for cameras with short focal length. But our interest, really, was not to calculate e. Our interest was to show the change in x with the latitude observing directly photo 1.
(This photo 1 is a mosaic made with real photos expanded or reduced to make comparable the original photos taken with different objectives. This mosaic is presented for the didactical objective of this paper. The measurements taken from observation of the photographs were made from the original photos, to avoid errors.)
It is a current saying "One image is worth more than one thousand words". Why not use that to teach Astronomy at school?!!!.
It is a good idea to suggest to interested teachers that they repeat the same experiment with their students. It is only necessary to take a photo from the same place (it is possible to use a special landmark: a post, a pillar, a tree, a corner of a house, a window...) with a common camera whose focal length is all that is necessary to know, and with a 100 ASA film. After that, when you have the set of photos (two photos are enough: an equinox and a solstice, but it is more beautiful to have three photos: an equinox and two solstices), you put one adjacent to the other one to form a continuous horizon (fig 5). You should be careful to connect the horizon correctly, with the minimum error. Then you measure and start the process.
Figure 6 shows the inclination of the Sun´s path over the horizon is 90ºf, the colatitude. This relationship can be verified by means of photography. It is also easy to compare results to stars' traces for different latitudes with the inclination of solar path. It is very easy to take some photos of the Sun near the sunrise or sunset point to determine the inclination of its path on the horizon. It can also be interesting to photograph the stars' traces near the east or west cardinal point. It is only necessary to put a reflex camera welllevelled on a tripod, so that the lower edge of the negative is parallel to the horizon. To take these kind of photos it is necessary to use a cable release, a sensitive film (slides 1600 ASA) and to expose the film for some minutes. It is a simple exercise to measure, on the photo, the angle determined by the star's trace or the line by the Sun's centres relative to the lower edge of the photo. This value should be the colatitude of the observation place.
If you carry out this experiment, please send me your group of photos at "ros@mat.upc.es". Good luck.
By taking measurements in degrees from the photos it is necessary to obtain the measurement d in cm and apply a mathematical relationship to convert d to x, which is expressed in degrees. To obtain the relationship between the distance d on the photo and the real angular distance x, the following details have been considered. The negative of the camera has a width of 36mm, so the maximum angular width of the camera b verifies, tan b/2 = 18/f because the camera, under an angle b/2, sees 18mm from the focal length f (fig 7).
Therefore the maximum angular width is
It is essential to consider that if a paper photo of 10x15 is used, as in our case, some corrections must be introduced. Really only 33mm of the negative appears on the paper copy (which has a width of 15.1 cm). By using a proportion, the corresponding distance to the 36 mm is 16.5 cm, and also this is the corresponding distance to the maximum angle b. Then the relationship between angles and distances is x/b = d/16.5 and the following formula is obtained,
The photos presented in this paper were taken with various cameras, with different focal lengths. In table II the focal length f, the distance d determined on the photos in cm and the angular distance x, obtained using the conversion explained above, appear.
Sunsets in Riga, Barcelona, Bogota, La Paz and Esquel. An equinox and two solstices have been published for each city.(Dates in table 2)
·Barcelona, Rosa M. Ros, Applied Mathematics 4, Technical University of Catalonia, Barcelona, Spain.
·Bogota, Juan Carlos Martínez, Observatorio Astronómico, Universidad Nacional de Colombia, Bogota, Colombia.
·La Paz, Gonzalo Pereira, Planetario Max Schreier, Universidad Mayor de S. Andrés, La Paz, Bolivia.
·Esquel, Néstor Camino, Complejo Plaza del Cielo, Universidad Nacional de la Patagonia, Esquel, Argentina.
·Gandia, España  Rosa Maria Ros
·Laponia, Finlandia  Sakari Ekko