Figure 1: Revolving Coordinate System for the Earth. Seasons and Latitude Simulation Step-by-Step

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1 1 z y x Figure 1: Revolving Coordinate System for the Earth Seasons and Latitude Simulation Step-by-Step The purpose of this unit is to build a simulation that will help us understand the role of latitude and the seasons for the Earth s climate and for the potential of solar energy. The key to building this simulation is understanding several different coordinate systems. The first coordinate system is shown in Figure 1. This coordinate system is attached to the Earth and is used to describe points on the Earth. It is useful for travel on the Earth. The origin is at the center of the Earth. The z-axis goes through the two poles with the positive z-axis in the direction of the North Pole. The positive x-axis is perpendicular to the z-axis and goes through the point where the prime meridian crosses the equator. The positive y-axis goes through the point on the equator with longitude 90 East. In Figure 1 the positive y-axis is going behind the paper. As the Earth revolves about its axis this coordinate system revolves with it. We normally use latitude and longitude to describe points on the Earth. Points on the equator have latitude 0 and points in the northern hemisphere have latitudes between 0 and 90 N. Points in the southern hemisphere have latitudes between 90 S and zero. The North Pole has latitude 90 N and the South Pole has latitude 90 S. In this unit we will use the notation φ for latitude and we will use radians rather than degrees. For points on the equator φ = 0; for points in the southern hemisphere π/2 φ < 0; and for points in the northern hemisphere 0 < φ π/2.

2 2 Points along the prime meridian have longitude 0. Points in the western hemisphere have longitudes between 0 and 180 W and points in the eastern hemisphere have longitudes between 0 and 180 E. In this unit we will use the notation θ for longitude and we will use radians rather than degrees. For points on the prime meridian θ = 0; for points in the western hemisphere π < θ < 0; and for points in the eastern hemisphere 0 < θ < π. Using this notation for latitude and longitude we can compute the coordinates of a point in our first coordinate system by x = R cos θ cos φ y = R sin θ cos φ z = R sin φ where R denotes the Earth s radius. In vector notation we write this u = R cos θ cos φ, sin θ cos φ, sin φ Question 1 Use the formula above to find the coordinates of the North Pole, the South Pole, and the point where the prime meridian crosses the equator. You can also find these coordinates directly based on the description of the coordinate system. Check that the answers agree. Question 2 Use the formula above to find the coordinates of the point whose latitude is 45 N on the prime meridian. Question 3 Use the formula above to find the coordinates of the place where you live. Question 4 Use the formula above to find a formula for the coordinates of the point whose latitude is φ on the prime meridian. The second coordinate system is shown in Figure 2. Its origin is at the center of the Earth and its z-axis goes through the two poles with the positive z-axis in the direction of the North Pole. The x- and y-axes however are fixed in space and do not revolve as

3 3 Figure 2: Two Frames in a Simulation in a Fixed Coordinate System for the Earth the Earth revolves. At time t = 0 this coordinate system coincides with the first one but as time changes the two coordinate systems coincide only once each day. If u denotes a position on the Earth in the first coordinate system and v(t) denotes the same point in the second coordinate system at time t measured in hours then v(t) = cos ( 2πt 24 sin ( 2πt 24 ) ( sin 2πt ) 24 0 ) ) cos ( 2πt u, where the vectors are written as column vectors. Notice that in the second coordinate system points on the Earth are rotating around the Earth s axis once every 24 hours and that the coordinates v(t) depend on time. Coordinates u in the first coordinate system do not depend on time. Question 5 Find the coordinates v(t) in the second coordinate system of the North Pole, the South Pole, and the point where the prime meridian crosses the equator. Question 6 Find the coordinates v(t) in the second coordinate system of the point whose latitude is 45 N on the prime meridian. Question 7 Find the coordinates v(t) in the second coordinate system of the place where you live.

4 4 z y x Figure 3: Tilting the Earth Question 8 Find a formula for the coordinates v(t) in the second coordinate system of the point whose latitude is φ on the prime meridian. The third coordinate system is shown in Figure 3 and has two similarities to the second one. Its origin is at the center of the Earth and it is fixed in space. It does not revolve as the Earth revolves. The xy-plane, however, is not the plane of the equator. Instead it is the plane of the ecliptic that is, the plane containing the Earth s orbit around the Sun. The angle between these two planes is 23.4, or radians. If v(t) represents a point on the Earth in the second coordinate system then this same point in the third coordinate system is represented by w(t) = cos(0.408) 0 sin(0.408) sin(0.408) 0 cos(0.408) v(t). Question 9 Find the coordinates w(t) in the third coordinate system of the North Pole, the South Pole, and the point where the prime meridian crosses the equator. Question 10 Find the coordinates w(t) in the third coordinate system of the point whose latitude is 45 N on the prime meridian. Question 11 Find the coordinates w(t) in the third coordinate system of the place where you live.

5 5 N θ S θ Figure 4: The Sun s angle and energy received Question 12 Find a formula for the coordinates w(t) in the third coordinate system of the point whose latitude is φ on the prime meridian. In this third coordinate system the Sun appears to rotate about the Earth. measure time in years, then its position at time s is given by the vector If we 150, 000, 000 cos(2πs), sin(2πs), 0, where we approximate the Earth s orbit around the Sun by a circle of radius 150,000,000 km. Figure 4 shows the basic reason why it is colder near the poles than near the equator and colder in winter than in the summer the rate at which a particular point on the Earth receives energy per square meter from the Sun depends on the angle of the incoming Sun s rays. When the Sun is directly overhead the rate of energy received per square meter is much higher than when the Sun is low in the sky. In Figure 4 the thick red line represents the constant rate at which energy ( watts 1 per square meter) from the Sun reaches the orbit of the Earth. The thick blue line represents the area over which this energy is spread because the energy is arriving at the Earth s surface at an angle. Looking at the triangle formed by the three thick lines we see that the rate at which a point on the Earth receives energy from the Sun is 1 Recall that a watt is a joule per second cos θ watts per square meter.

6 6 Two unit vectors are important the vector N, called the normal vector, that is perpendicular to the Earth s surface at the point of interest and the vector S that points from the point of interest toward the Sun. Using these two vectors the incoming energy rate is ( N S) watts per square meter. if the Sun is above the horizon. If the Sun is below the horizon then N S would be negative, so this formula must be modified to max(0, N S) watts per square meter. It will be convenient throughout this unit to write vectors in magnitude-direction form that is, in the form where p is the unit vector q = q p 1 q q. Now that we ve done the necessary mathematical work, we are ready to build a series of simulations that can help us understand the effects of latitude and time of year on climate. Begin by launching DIYModeling and opening the file seasons-skeleton.xml. Look at the components tab in the model editor. You should see something like the top screenshot in Figure 5. Notice there are four components in addition to a camera The Earth, the Sun, and two arrows. The light in this simulation comes from the Sun. You can see the four components other than the camera in the simulation in the bottom screenshot in the same figure. Each of these components has key parameters or attributes that are driven by an underlying model. You will modify the underlying model in this unit. See Figure 6. Earth Primary driven by the variable northpole Secondary driven by the variable equator

7 Figure 5: Components 7

8 8 Figure 6: The Model Tab

9 9 Sun Position driven by * sun this places the Sun 150,000,000 kilometers from the Earth. NorthPoleArrow Size [1000, 1000, 1000] Color Chosen using the color picker. ArrowTail driven by 6371 * northpole ArrowHead driven by 8500 * northpole PrimeMeridianArrow Size [1000, 1000, 1000] Color Chosen using the color picker. ArrowTail driven by 6371 * equator ArrowHead driven by 8500 * equator The way that the Earth is placed in a simulation is controlled by the two parameters Primary and Secondary. Each of these should be unit vectors and they should be perpendicular to each other. The first parameter Primary is the direction in which the North Pole is pointing and the second parameter Secondary is the direction in which the point where the prime meridian crosses the equator is pointing. Notice that the tail of each arrow is located on the Earth s surface 6,371 kilometers from the Earth s center and the head of each arrow is located 8,500 kilometers from the Earth s center. The direction of each arrow is determined by the appropriate variable. Question 13 As the Earth revolves about its axis in 24 hours in what direction should the point where the prime meridian crosses the equator be pointing? Check your answer by modifying the definition of the variable equator Question 14 The next step is to tilt the Earth as described above. In what direction should the North Pole be pointing? In what direction should the point where the prime meridian crosses the equator be pointing? Check your answers by modifying the definition of the variables northpole and equator

10 10 Now we d like to look at the season, or time of year. In the components tab drag a slidercontrol component from the Components Library pane into the pane with the other components. Click the name of this component and change it to season. This component has the following parameters. Label Edit this to read Season this is the label that will appear with this slider control. Value Edit this to read 0 This is the value of this slider when the simulation starts. Minimum Edit this to read 0 This slider can have values between 0 and 1. Maximum Edit this to read 1 This slider can have values between 0 and 1. Choose Fixed-Point and 2 in the two pull-down menus for the format. Now you can use the variable season in the model to control the position of the Sun. Question 15 Edit the expression for the variable sun in the Model tab to control the position of the Sun and, thus, the seasons. Check your work by running the new simulation. Play with this simulation to see the effects of the seasons. What setting of the season slider control corresponds to winter in the northern hemisphere? What settings of the slider control correspond to the two equinoxes? Our simulation so far is good for visualizing the effects of latitude and season but we d like to look at some of the numbers. Add two new digitaldisplay components to the simulation. Name one of them NorthPoleMeter and the other EquatorMeter. These components each have several parameters. Label This parameter specifies the label that identifies a digital display in the simulation. Format Choose Fixed-Point and 2 from the pull-down menus. Value This parameter is an expression specifying the reading that appears in the digital display. For example, * max(0, dot(sun, equator))

11 11 displays the rate in watts per square meter at which energy is reaching the Earth at the point where the prime meridian intersects the equator. Question 16 Modify the simulation to display the rate in watts per square meter at which energy is reaching the Earth at the North Pole and at the point where the prime meridian crosses the equator. So far the simulation allows us to compare only a point at the North Pole and a point on the equator. The next question adds additional capability. Question 17 Add another variable to the model for a point at a latitude chosen by the player. Add a slider control that enables the player to choose the latitude (in radians), an arrow that marks the point chosen by the player in the same way that arrows mark the other two points, and a digital display that displays the rate per square meter at which the new point is receiving energy from the Sun. Next we want to add one more feature to this simulation that will enable us to find the total amount of energy, measured in Watt hours per square meter, coming from the Sun over the course of one day at different latitudes at different times of the year. As your simulation runs, your meters are displaying the rate at which energy is coming in max(0, N S) at each point and time of day during a particular season. The integral max(0, N S) dt computes the number of Watt hours of energy per square meter over the course of one day at the point and season. This is exactly the number we want and it is easy to compute in DIYModeling. We simply define a new variable q(t) by the initial value problem Notice that the integral we want is dq dt = max(0, N S), q(0) = 0.

12 max(0, N S) dt = q(24). As your simulation runs in DIYModeling it can compute q(t). We need to stop the computation when t reaches 24. This requires a slight modification. We define { dq dt = 1367 max(0, N S), if t 24; 0, if 24 < t. The syntax in DIYModeling for an expression like the one on the right side of the differential equation above is 1367 * If(t <= 24, max(0, dot(choice, sun)), 0) where choice is the name you chose for the variable representing N and sun is the name you chose for the variable representing S. Figure 7 shows a screen shot of the Model tab in the Model Editor as you add a new variable. Notice the arrows corresponding to the following steps. 1. Add a new variable by clicking the Add New button. 2. Enter the name of the new variable in the Name column. 3. Choose Diff Eq from the pull down list in the How column. 4. Choose Decimal from the pull down list in the What column. 5. Enter the initial value in the Initial Value column. 6. Enter the expression on the right hand side of the differential equation in the Expression column. Figure 8 shows a simulation that should look similar to the one you build now using the ideas we just discussed. Use this simulation to answer the following questions. Question 18 What is the total amount of energy in Watt hours per square meter coming from the Sun at the North Pole on the day of the summer solstice in the Northern Hemisphere?

13 13 Figure 7: Adding a New Differential Equation Variable Figure 8: Daily Energy

14 14 Question 19 What is the total amount of energy in Watt hours per square meter coming from the Sun on the equator on the day of the summer solstice in the Northern Hemisphere? Question 20 What is the total amount of energy in Watt hours per square meter coming from the Sun at latitude 45 N on the day of the summer solstice in the Northern Hemisphere? Question 21 What latitude receives the greatest total amount of energy on the day of the summer solstice in the Northern Hemisphere? We can take this simulation a step further computing the annual energy received at a given point on the Earth. Now we want to make the Sun rotate around the Earth in 365 days. We ll use the vector S(t) = ( ) ( ) 2πt 2πt cos, sin, 0, since we re measuring time in hours, and the integral we need is max(0, N S) dt. There is one small problem. The simulation we ve built thus far is designed for studying what happens over the course of a day. It will run too slowly for simulating what happens over the course of a year. In the Project tab of the Model Editor you can change the speed of the simulation. See Figure 9. Edit the Time Scale box to read: 24@hour@/4@second@ This adjusts the speed of the simulation so that 24 hours in the simulated world unfold in 4 seconds in the real world. This is a good compromise. The simulation of a year will take a reasonable amount of time and the Earth won t be spinning too quickly.

15 Figure 9: Changing the Time Scale 15