ESCI 110: Earth-Sun Relationships Page 4-1. Exercise 4. Earth-Sun Relationships and Determining Latitude

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1 ESCI 110: Earth-Sun Relationships Page 4-1 Introduction Exercise 4 Earth-Sun Relationships and Determining Latitude As the earth revolves around the sun, the relation of the earth to the sun affects the seasons, length of day and night, position of the sun above the horizon and therefore directly affects many phenomena on the earth's surface such as climate, soils, agriculture and life style. In this exercise the student will examine some of the basic relationships between the earth and its nearest star, and learn how this information can be used to determine latitude on the surface of the earth. Objectives Upon completion of this exercise, you should be able to: Describe the conditions characteristic of each of the solstices and equinoxes and be able to-describe and explain the changing seasons. Explain the reversal of seasons that exists between the northern and southern hemispheres. Be able to explain the ever changing duration of daylight and why some areas experience 24 hours of daylight or darkness. Determine the latitude of a point knowing the date and the angle of the sun above the horizon. Find the latitude of a point by observation of the north star. Materials Two meter sticks Graph paper Protractor Background Information The earth rotates on its axis once per day, and this causes the sun to trace a path through the sky, rising at dawn and setting at dusk. In the middle of the day, the sun will appear to be at its highest point above the horizon, and the time will be solar noon. If you observe the position of the noon sun repeatedly throughout the year, you will find that its altitude above the horizon changes. In the winter, the sun is lower in the sky and shadows are longer, while in summer the sun is higher and shadows are shorter. This occurs because the earth is tilted on its axis by 23 1/2º relative to the plane of its orbit, and the relative position of the earth and sun changes throughout the year as the earth revolves around the sun.

2 ESCI 110: Earth-Sun Relationships Page 4-2 Activity The idea behind this exercise is to find a vertical object of known height, and measure the length of its shadow at noon. The easiest way to complete this activity is to use two meter sticks, and have one student stand one of the meter sticks so that it is vertical, while a second student measures the shadow. In fact, yard sticks will work, as will a measuring tape and a vertical fence post. In the Los Angles area solar noon is typically a few minutes before noon Pacific Standard Time, and a few minutes before 1 PM Pacific Daylight Savings Time. (This will change at other locations). At noon the shadow will point due north, and also will be the shortest at any time during the day. Once you have measured the height of the object and the length of the shadow, you are now ready to use graph paper to determine the sun s altitude (the angle between the sun position and the horizon). Mark a point in the lower left corner of the graph paper. Draw a vertical line upward to represent the height of the object to scale (if you used the meter sticks, use a scale of 10:1, where 10 centimeters on the stick is represented by one centimeter on your drawing). Now draw a horizontal line from the starting point toward the right on the graph paper to represent the length of the shadow (use the same scale as you did for the vertical object). You should now have two line segments that meet at a 90º angle, and can be used to form two sides of a triangle. Use a straight edge to connect the top of the meter stick on the diagram to the end of the shadow, and complete the triangle. Use a protractor to measure the angle between the ground (the shadow line) and the hypotenuse of the triangle. This angle is the altitude of the sun. In the space below, record the sun s altitude, and (critically) the date on which you made your measurements: Altitude: Date:

3 ESCI 110: Earth-Sun Relationships Page 4-3 Background Information Fig This is a diagram of Earth as viewed space. The Sun is located to the right. Note how the left side of Earth is shaded. This diagram is set for the Summer Solstice and shows several of the important imaginary lines on the surface of the Earth. First we have the Circle of Illumination. This line demarks the separation of the sun-illuminated side of the Earth from the side in darkness. This is the same line that you see at night on the Moon when you look at a crescent moon. Note that the Circle of Illumination divides the Earth into two hemisphere. No matter what the day or hour, on-half of the Earth is in darkness and one-half is in sunlight. Perpendicular to the Circle of Illumination is the Plane of the Ecliptic. This represents the equatorial plane of the Sun. Both of these lines are related to the position of the Sun or its direction. There are two similar lines that are related to the rotation of the Earth on its axis. First is the Axis of Rotation which passes through the North and South Poles of Rotation. Perpendicular to this is the Equator, which divides the Earth into a Northern and a southern Hemisphere. Note that in this diagram the Axis of Rotation is tilted 23 1/2 degrees from the Circle of Illumination. The Equator, which is perpendicular to the Axis of Rotation, is tilted 23 1/2 degrees from the Plane of the Ecliptic.

4 ESCI 110: Earth-Sun Relationships Page 4-4 Now let us look at how the Sun s rays intersect the Earth s surface. Since the Sun is approximately 93 million miles away, for all practical purposes its rays are all parallel by the time they reach the Earth. In Fig. 4-1 the Sun s rays are perpendicular to the Earth s surface at a point that is 23 1/2 degrees North Latitude (Point A), or the Tropic of Cancer. This is as far north as the Sun ever appears to be directly overhead at noon. This marks the Summer Solstice. Six months later at the Winter Solstice the Sun will be directly overhead at 23 1/2 degrees South Latitude. This marks the position of the Tropic of Capricorn. On the shown, the Summer Solstice, the Sun s rays are tangential to the surface of the Earth at 66 1/2 degrees North and South Latitude. At 66 1/2 degrees north Latitude, one can see that as the Earth rotates on its axis, all land north of that latitude will receive 24 hours of sunlight. The 66 1/2 degree North Latitude line is called the Arctic Circle. At the 66 1/2 degree line of South Latitude you can see that as the Earth rotates on its axis all of the land south of that latitude receives 24 hours a day of darkness. The 66 1/2 degree South Latitude line is called the Antarctic Circle. Six months later, at the Winter Solstice, the Sun s rays will be tangential at the same two points. However at this time the land above the Antarctic Circle will receive 24 hours a day of sunlight while the land above the Arctic Circle will be in darkness for 24 hours a day. All four of these lines, the tropics and the circles, are lines of latitude and are therefore parallel to the equator. We refer to lines of latitude as being parallels because of this relationship. By inspection of this diagram you can see that of all the possible parallels, only the equator is bisected equally by the Circle of Illumination. At the Equator there is an equal division between sunlight and darkness, or 12 hours of each. In fact, the Equator will receive 12 hours of sunlight and 12 hours of darkness every day of the year. It follows that by locating the Circle of Illumination on a diagram of the globe, one can determine the amount of daylight or darkness a particular spot will receive. Figure 4-2. Four positions of the earth relative to the sun. The large circle in the center represents the sun. The four small circles show the earth s position at four different times (times A-D). The tilt of the earth s axis is indicated by the lines marking the north (top) and south (bottom) poles.

5 ESCI 110: Earth-Sun Relationships Page 4-5 Questions 1. On Fig. 4-2 draw in the equator and circle of illumination on each earth, and fill in each date. 2. At what latitude are the Sun s rays perpendicular to the earth's surface at A? ; at B? ; at C? ; at D?. 3. How many hours of daylight are there north of 66.5 N. on Fig. I at A? South of 66.5 S on diagram A? ; At the equator on all diagrams. 4. What season is represented in the time period between the following: N. Hemisphere S. Hemisphere A & B ; ; B & C ; ; C & D ; ; D & A ; ; The position of the noon sun above the horizon is dependent on the day of the year and latitude at which you observe the position of the noon day sun. The following relationships always exist. Figure 4-3 The zenith is the point directly overhead at any location on the Earth s surface

6 ESCI 110: Earth-Sun Relationships Page 4-6 The zenith angle always equals the number of degrees between the line of sight to the noon sun and the zenith The altitude of the noon sun is the angle between the horizon and the line of sight to the noon sun The zenith angle always equals the number of degrees between the line of sight to the noon sun and the zenith. Using the definitions given above and referring to Fig. 4-3 we can derive an equation to relate these various angles as follows: Zenith Angle = 90 O Altitude of Noon Sun The form of this equation can be changes in order to solve for any of the other two quantities as follows: Altitude of Noon Sun = 90 O - Zenith Angle 90 O = Zenith Angle + Altitude of Noon Sun Throughout the year the sun traces a path on the earth s surface where it is directly overhead in the sky. The point at which the sun is directly overhead is called the sun position or the Latitude of Noon Sun (LS). The sun position on any given date can easily be determined from a graph called an analema. Figure 4-4 below is a copy of the analema. The vertical scale gives the latitude of the sun position for a particular day (called the declination). From the analema, what is the latitude of the sun position (or declination) for September 5?

7 ESCI 110: Earth-Sun Relationships Page 4-7 Figure 4-4. The analema. Use the analema to determine the sun position for the date on which you measured the sun s altitude:

8 ESCI 110: Earth-Sun Relationships Page 4-8 Fig How to calculate the altitude of the noon day sun. Fig. 4-5 is an expansion on Fig. 4-3 in that the line marking the Zenith has been extended to the center of the Earth forming line BC. There the line marked BC intersects a line taken from where the Sun is directly overhead at noon, line AC. The intersection of these two lines creates an angle that is the same as the zenith angle that you would observe at noon. You can prove by geometry that these two angles are the same. HINT: you have two parallel lines intersected by another line. Inspecting this diagram shows that the Zenith angle that you observed is the same as the sum of the latitude of your location, Latitude of Place (LP), and the latitude where the sun is directly overhead at noon, Latitude of Sun (LS), as measured from the center of the Earth. This is true because as, shown in the diagram, you are located in one hemisphere and the sun is located in the other hemisphere. We can express this relationship as an equation as follows: Zenith Angle = LP + LS

9 ESCI 110: Earth-Sun Relationships Page 4-9 We have already express a relationship between the zenith angle and the altitude of the noon day sun. We can use these two equations to calculate the altitude of the noon day sun as follows: Question: You are located at 20 O N on December 22. What would be the altitude of the noon day sun on that day and at your location? Solution: You will need both of the equations derived above to solve this problem. First we must determine the Zenith angle. Since we are looking for the altitude of the noon day sun we cannot start with that equation. But we can use the LP + LS equation as follows: Zenith angle = LP + LS Zenith angle = 20 O N {given to us} O S {determined from the analema} Draw out on board Zenith angle = 43.5 O Knowing the zenith angle we can now solve for the altitude of the noon day sun. Altitude = 90 O Zenith angle Altitude = 90 O 43.5 O Altitude = 46.5 O 5. Solve the following problems. a. It is Mar. 21. An observer is located at 50 S. What is the altitude of the sun above the horizon at noon?

10 ESCI 110: Earth-Sun Relationships Page 4-10 b. It is Mar. 21. An observer is located at 26 N. What is the altitude of the noon sun? c. It is June 22. An observer is located at 60 S. What is the altitude of the noon sun? d. It is June 22. An observer is located at 2 N. What is the altitude of the noon sun? e. It is Dec. 22. An observer is located at 80 N. What is the altitude of the noon sun?

11 ESCI 110: Earth-Sun Relationships Page 4-11 f. It is April 1. An observer is located at 40 S. What is the altitude of the noon sun? How do we know? Fig Locating your latitude if you and the sun are located in the same hemisphere. Fig. 4-6 is similar to Fig. 4-5 except that now the sun and you are located in the same hemisphere. We still have the same relationship between the various angles. However, now the zenith angle measured from the center of the Earth is determined by subtracting the smaller of LP and LS form the larger. Since LS can never be larger than 23.5, we usually write the equation as:

12 ESCI 110: Earth-Sun Relationships Page 4-12 Zenith angle = LP LS Problem: It is noon on June 22 at the observer s location. The Sun is 30 above the southern horizon. What is the latitude of the observer? Solution: We know the altitude of the noon day sun so we may start with Altitude = 90 Zenith Angle Or Zenith Angle = 90 - Altitude Zenith Angle = Zenith Angle = 60 From the analema we can determine the LS as being 23.5 N. We can solve for LP as follows: Zenith Angle = LP ± LS Remember that we use the + when the two are in different hemispheres and the when they are in the same hemisphere. This means that there are two possible solutions as follows: 60 = LP N LP = LP = 36.5 S Or 60 = LP 23.5 N LP = LP = 83.5 N

13 ESCI 110: Earth-Sun Relationships Page 4-13 To determine which of these is correct, they both can t be, we must look for more information in the problem. The problem states that the Sun is over the southern horizon. This means that you must be north of the location of the Sun. Only one of the two possible locations determined above has the observer north of the sun and that is the second location. Therefore: LP = 83.5 N This is also logical since the Sun will be very low in the sky when you are at 83.5 latitude, either N or S. 6. Solve the following problems. a. It is noon on June 22 at an observer's location. The sun is 10º above the northern horizon. What is the latitude-at his location? b. It is noon on Sept. 22 at an observer's location. The sun-is 75º above the northern horizon. What is the observer s latitude? c. The sun is 65 above the southern horizon. It is Dec. 22. What is the latitude of the observer?

14 ESCI 110: Earth-Sun Relationships Page 4-14 d. The sun is directly overhead at 10 N. The altitude of the sun at an observer's location is 75 above the northern horizon. What is his latitude? e. The sun is overhead at 20 S. The attitude of the sun at an observers location is 10 above the southern horizon. What is his latitude? f. Now you can use the correct formula and calculate the zenith angle at the location you used in the first activity in this exercise. 8. If you are located at the following latitudes how high will the north star be above the horizon? 0 15 N. 25 N. 70 N. 90 N. 15 S.

15 ESCI 110: Earth-Sun Relationships Page 4-15 Can the North Star be used for navigation in the southern hemisphere? Key Words to Define Write the definition of the following key words. Axis of rotation Circle of illumination Rotation Solar noon Revolution Analema Tropic of Cancer Tropic of Capricorn Arctic Circle Antarctic Circle Equator Plane of the ecliptic