Heavenly Mathematics: Cultural Astronomy

Transcription

1 GEK 1506 Heavenly Mathematics: Cultural Astronomy SUN HOMEWORK Done by: Toh Yong Chen Alex Choi Mun Tow Sim Wei Ting Neo Su Zhen Lim Luo Bing U030610A U044259Y U044282H U054229J U054303W Group : 18 Course Tutor: Helmer Aslaksen

2 Objective The aims of this homework are to observe the rising positions of the Sun over three months, attempt to determine the date of the fall equinox and study the shape of the shadow path of a vertical sundial. Observations of Sunrise Positions E 30 th August hours 1) The azimuth of the sun is approximately 75 degrees which is slightly North of East. Picture 1 E 23rd September hours 2) The azimuth of the sun is approximately 90 degrees which is directly due East. Picture 2 E 22 nd October hours 3) The azimuth of the sun is approximately 110 degrees which is slightly South of East. Picture 3 1

3 Explanation for the Sun s rising position Clearly, in the course of 3 months, the rising position of the sun has shifted. The apparent yearly path of the sun is known as the ecliptic. This circular path is tilted 23.5 degrees relative to the celestial equator because the Earth s rotation axis is tilted by 23.5 degrees with respect to its orbital plane. Ecliptic Plane N August observation Celestial Equator September Equinox October observation The above diagram shows the positions of important markers and our observations. N S 2

4 Rotating the first diagram, we get the above view that reflects our position in Singapore (near equator) as well as the path of the Sun on the respective month as viewed from the horizon. The yellow dashed ellipse represents the path of the Sun through the day of our observation in August and the orange dashed ellipse represents the path of the Sun through the day of our observation in October. During the September Equinox, the path of the Sun follows the Celestial Equator. Before the September equinox (occurring around 23 rd September), the ecliptic plane is above the Celestial Equator when viewed from North. The sun will rise in the slightly North of East and will set slight North of West. On the day of the September Equinox, the ecliptic intersects the celestial equator and it will rise direct East and will set direct West. After the September equinox, the ecliptic plane is below the celestial equator when viewed from north. Then the sun will rise slightly South of East and will set slightly South of West. 3

5 Ring Observations A ring was suspended in the East-West plane by nylon strings held taut by one group member at each end. The observations were done around the same time of the day in the three months so that a clear contrast can be made. For all the ring observations, the red dashed arrow in the photographs points to the shadow of the bottom half of the ring while the blue solid arrow points to the shadow of the top half of the ring. 27 th August hours The shadow of the ring was cast South of the ring when viewed from the top and followed a ring shape. The shadow of the top half of the ring did not coincide with the bottom half of the ring. 22 nd September hours and 23 rd September hours In both days, the shadow was cast directly below the ring and the shape was in a solid rectangle. This implies that the shadow of the top half of the ring coincided with the bottom half of the ring. 4

6 21 st October hours The shadow was cast North of the ring when viewed from the top and followed a ring shape. The shadow of the top half of the ring did not coincide with the bottom half of the ring. Explanation for the ring observations As explained earlier in the sunrise observations, we know that the path of the sun through the day in Singapore is different in the three months. In August, the Sun rises slightly North of East and sets slightly of North of West. The path of the Sun through the day is thus always on the North side. As such, the shadow of the ring will be cast on the South of our ring. On 23 September, the autumnal equinox occurred and the Sun rises direct east crosses the meridian at the zenith and sets direct West. As a result, we get a solid rectangular shadow. In October, the Sun rises slightly South of East and sets slightly of South of West. The path of the Sun through the day is thus always on the South side. As such, the shadow of the ring will be cast on the North side of our ring. The explanation is similar to the stick observations. 5

7 Stick Observations A stick was fastened in the middle of white sheets of paper with blue-tac. Red sticky tape marked the position of the shadows at approximately 2 hour intervals between 0900 hours to 1700 hours. At the end of the day, we had 5 readings and we traced the path of the shadows with a black marker. 27 th August 2005 Stick Observations Reading 1 Reading 2 Reading 3 Reading 4 Reading 5 All the shadows fell on the South side of the stick with varying lengths at different times of the day. The path of shadows followed a slight concave arc south of the stick. 6

8 22 nd September 2005 Stick Observations Reading 1 Reading 2 shadow Reading 3 Reading 4 Reading 5 Instead of the chopstick used in August, we used a longer stick for our observations as we found out that a longer stick will be able to amplify the curvature of the shadows cast by the sun. In September, the shadows formed a straight line on the East-West plane. 7

9 21 st October 2005 Stick Observations Reading 1 Reading 2 Reading 3 Reading 4 Reading 5 All the shadows fell on the North side of the stick with varying lengths at different times of the day. The path of shadows followed a concave arc north of the stick. 8

10 Explanations of stick observations Sun Aug Sept Oct Sunlight August N Stick S Ground Shadows The above diagram illustrates why the shadows of a particular day all fall at a September October certain side of the stick. In August, the Sun s path through the day is always North. As such, the shadows will be cast South of our stick. In September, the Sun rises direct east crosses the meridian at the zenith and sets direct West. As a result, we get a vertical line in the East-West plane. In October, the Sun s path through the day is always South. As such, the shadows will be cast North of our stick. 9

11 Sun at higher altitude Sun at lower altitude Length of shadow for Sun at lower altitude Ground Stick Length of shadow for Sun at higher altitude The above diagram shows that there is an inverse relationship between the altitude of the sun and the length of the shadow of the stick formed. The higher the altitude is, the shorter the shadow of the stick. Therefore the shadow of the stick will be longer in the dawn and dusk as compared to the shadow of the stick for the rest of the day. As the path of the Sun rises from low altitude during dawn to high altitude during noon and then down to low altitude again during dusk, we will get a concave curve as long as the Sun is not directly over the East-West plane. An overview of the paths of the shadows Path of shadow in September Stick Path of shadow in October W E Top view of shadows Path of shadow in August 10

12 Insights and Reflection During the course of our Sun Homework, many interesting incidents took place that helped to make this homework a memorable one. As we had to take 5 shadow readings at different times of the day, we were very much at the mercy of the weather for the days. There were times when we had taken photos of the shadows till noon and it started raining. As a result, we had to repeat the entire process to obtain the readings some other day. During a trip down to the photo-taking site (which was beside a construction site near a group mate s house), there was once when the strong winds blew our entire set-up away! One of our group members thought we played a prank on him. Only upon scouring the entire area did he realize it was gusts that blew it away. We later used our bags/shoes/stones/textbooks (that were meant for our revision) to hold the set-up in its place. Besides this, inquisitive policemen, as well as the construction site personnel, also came to question our intentions for hanging around the area for such a long period of time. When we initially started with our homework, we found that the length of the shadow cast by our miserable chopstick was not significant enough to indicate the curvature of the shadow. We later discovered that we should use a longer stick to amplify the curvature of the shadow path. 11

13 As for the ring assignment, initially, we did not know how to suspend the ring such that we would be able to take a good picture of its shadow without including the shadows of the objects that was used to suspend the ring. However, one of our brilliant group members later suggested using nylon strings to suspend the ring so that we would be able to take pictures of its shadow as the shadow of the nylon strings will be less obvious. However another problem arose. When the wind blew, the ring will be shaking and we had to wait for the ring to stabilize (and our hands to be firm) before we could take the picture. This requires team co-operation and good timing. One of the group members has always wondered why her father would park the bike at different positions at different times of the year. After the Sun homework, we are now able to understand the rationale for doing so. The photo below is a view of the car park taken from her kitchen s window. As the photo was taken during October, the Sun was on the South side and as a result, only the building on the left would cast a shadow on the left side (as viewed in the picture) of the car park. However during March equinox to September equinox, the Sun will be in the North and only the building on the right will cast a shadow on the right side of the car park. Therefore in order not to let the Sun heat up the bike s seat, her father will park his bike on the left hand side of the car park from September equinox to March N equinox and on the right side of the car park from March equinox to September equinox. Overall, we would say that all these events we have encountered have facilitated our understanding of this module and its applications in real life, as well as building our team cohesiveness. 12