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z

x

y

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 understandthe role of latitude and the seasons for the Earth’s climate and for the potentialof solar energy.

The key to building this simulation is understanding several different coordinate sys-tems. The first coordinate system is shown in Figure 1. This coordinate system is attachedto 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 thepositive z-axis in the direction of the North Pole. The positive x-axis is perpendicular tothe z-axis and goes through the point where the prime meridian crosses the equator. Thepositive y-axis goes through the point on the equator with longitude 90◦ East. In Figure 1the positive y-axis is going behind the paper. As the Earth revolves about its axis thiscoordinate system revolves with it.

We normally use latitude and longitude to describe points on the Earth. Points on theequator have latitude 0 and points in the northern hemisphere have latitudes between 0and 90◦ N. Points in the southern hemisphere have latitudes between 90◦ S and zero. TheNorth 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 ratherthan 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.

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Points along the prime meridian have longitude 0. Points in the western hemispherehave longitudes between 0 and 180◦ W and points in the eastern hemisphere have longitudesbetween 0 and 180◦ E.

In this unit we will use the notation θ for longitude and we will use radians rather thandegrees. 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 apoint in our first coordinate system by

x = R cos θ cosφy = R sin θ cosφz = R sinφ

where R denotes the Earth’s radius. In vector notation we write this

~u = R 〈cos θ cosφ, sin θ cosφ, sinφ〉

Question 1 Use the formula above to find the coordinates of the North Pole, the SouthPole, and the point where the prime meridian crosses the equator. You can also findthese coordinates directly based on the description of the coordinate system. Check that theanswers agree.

Question 2 Use the formula above to find the coordinates of the point whose latitude is45◦ 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 whoselatitude is φ on the prime meridian.

The second coordinate system is shown in Figure 2. Its origin is at the center of theEarth and its z-axis goes through the two poles with the positive z-axis in the directionof the North Pole. The x- and y-axes however are fixed in space and do not revolve as

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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 butas time changes the two coordinate systems coincide only once each day. If ~u denotes aposition on the Earth in the first coordinate system and ~v(t) denotes the same point in thesecond coordinate system at time t measured in hours then

~v(t) =

cos(

2πt24

)− sin

(2πt24

)0

sin(

2πt24

)cos(

2πt24

)0

0 0 1

~u,

where the vectors are written as column vectors. Notice that in the second coordinatesystem points on the Earth are rotating around the Earth’s axis once every 24 hours andthat the coordinates ~v(t) depend on time. Coordinates ~u in the first coordinate system donot 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 whoselatitude is 45◦ N on the prime meridian.

Question 7 Find the coordinates ~v(t) in the second coordinate system of the place whereyou live.

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z

x

y

Figure 3: Tilting the Earth

Question 8 Find a formula for the coordinates ~v(t) in the second coordinate system ofthe point whose latitude is φ on the prime meridian.

The third coordinate system is shown in Figure 3 and has two similarities to the secondone. Its origin is at the center of the Earth and it is fixed in space. It does not revolve asthe Earth revolves. The xy-plane, however, is not the plane of the equator. Instead it isthe 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 0.408 radians. If ~v(t) represents a pointon the Earth in the second coordinate system then this same point in the third coordinatesystem is represented by

~w(t) =

cos(0.408) 0 sin(0.408)0 1 0

− 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 whoselatitude is 45◦ N on the prime meridian.

Question 11 Find the coordinates ~w(t) in the third coordinate system of the place whereyou live.

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θ

θ

~N

~S

Figure 4: The Sun’s angle and energy received

Question 12 Find a formula for the coordinates ~w(t) in the third coordinate system ofthe point whose latitude is φ on the prime meridian.

In this third coordinate system the Sun appears to rotate about the Earth. If wemeasure time in years, then its position at time s is given by the vector

150, 000, 000 〈cos(2πs), sin(2πs), 0〉,

where we approximate the Earth’s orbit around the Sun by a circle of radius 150,000,000km.

Figure 4 shows the basic reason why it is colder near the poles than near the equatorand colder in winter than in the summer – the rate at which a particular point on theEarth receives energy per square meter from the Sun depends on the angle of the incomingSun’s rays. When the Sun is directly overhead the rate of energy received per square meteris much higher than when the Sun is “low in the sky.” In Figure 4 the thick red linerepresents the constant rate at which energy (1367.6 watts1 per square meter) from theSun reaches the orbit of the Earth. The thick blue line represents the area over which thisenergy is spread because the energy is arriving at the Earth’s surface at an angle. Lookingat the triangle formed by the three thick lines we see that the rate at which a point on theEarth receives energy from the Sun is

1367.6 cos θ watts per square meter.

1Recall that a watt is a joule per second.

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Two unit vectors are important – the vector ~N , called the normal vector, that isperpendicular to the Earth’s surface at the point of interest and the vector ~S that pointsfrom the point of interest toward the Sun. Using these two vectors the incoming energyrate is

1367.6( ~N · ~S) watts per square meter.

if the Sun is above the horizon. If the Sun is below the horizon then ~N · ~S would benegative, so this formula must be modified to

1367.6 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

~q = ||~q||~p

where ~p is the unit vector

1||~q||

~q.

Now that we’ve done the necessary mathematical work, we are ready to build a series ofsimulations 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 atthe components tab in the model editor. You should see something like the top screenshotin Figure 5. Notice there are four components in addition to a camera – The Earth, theSun, and two arrows. The light in this simulation comes from the Sun. You can see thefour components other than the camera in the simulation in the bottom screenshot in thesame figure.

Each of these components has key parameters or attributes that are driven by anunderlying 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

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Figure 5: Components

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Figure 6: The Model Tab

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• Sun

◦ Position – driven by 150000000 * sun – this places the Sun 150,000,000 kilo-meters 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 parametersPrimary and Secondary. Each of these should be unit vectors and they should be per-pendicular to each other. The first parameter Primary is the direction in which the NorthPole is pointing and the second parameter Secondary is the direction in which the pointwhere the prime meridian crosses the equator is pointing.

Notice that the tail of each arrow is located on the Earth’s surface – 6,371 kilometersfrom the Earth’s center – and the head of each arrow is located 8,500 kilometers from theEarth’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 shouldthe point where the prime meridian crosses the equator be pointing? Check your answer bymodifying the definition of the variable equator

Question 14 The next step is to tilt the Earth as described above. In what direction shouldthe North Pole be pointing? In what direction should the point where the prime meridiancrosses the equator be pointing? Check your answers by modifying the definition of thevariables northPole and equator

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Now we’d like to look at the season, or time of year. In the components tab drag asliderControl component from the Components Library pane into the pane with the othercomponents. Click the name of this component and change it to season. This componenthas the following parameters.

• Label – Edit this to read Season – this is the label that will appear with this slidercontrol.

• Value – Edit this to read 0 – This is the value of this slider when the simulationstarts.

• 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 theposition 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 slidercontrol corresponds to winter in the northern hemisphere? What settings of the slidercontrol correspond to the two equinoxes?

Our simulation so far is good for visualizing the effects of latitude and season butwe’d like to look at some of the numbers. Add two new digitalDisplay components tothe simulation. Name one of them NorthPoleMeter and the other EquatorMeter. Thesecomponents each have several parameters.

• Label – This parameter specifies the label that identifies a digital display in thesimulation.

• Format – Choose Fixed-Point and 2 from the pull-down menus.

• Value – This parameter is an expression specifying the reading that appears in thedigital display. For example,

1367.6 * max(0, dot(sun, equator))

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displays the rate in watts per square meter at which energy is reaching the Earth atthe point where the prime meridian intersects the equator.

Question 16 Modify the simulation to display the rate in watts per square meter at whichenergy is reaching the Earth at the North Pole and at the point where the prime meridiancrosses the equator.

So far the simulation allows us to compare only a point at the North Pole and a pointon the equator. The next question adds additional capability.

Question 17 Add another variable to the model for a point at a latitude chosen by theplayer. Add a slider control that enables the player to choose the latitude (in radians), anarrow that marks the point chosen by the player in the same way that arrows mark theother two points, and a digital display that displays the rate per square meter at which thenew point is receiving energy from the Sun.

Next we want to add one more feature to this simulation that will enable us to findthe total amount of energy, measured in Watt hours per square meter, coming from theSun over the course of one day at different latitudes at different times of the year. As yoursimulation runs, your meters are displaying the rate at which energy is coming in

1367.6 max(0, ~N · ~S)

at each point and time of day during a particular season. The integral

1367.6∫ 24

0max(0, ~N · ~S) dt

computes the number of Watt hours of energy per square meter over the course of one dayat the point and season. This is exactly the number we want and it is easy to compute inDIYModeling. We simply define a new variable q(t) by the initial value problem

dq

dt= 1367.6 max(0, ~N · ~S), q(0) = 0.

Notice that the integral we want is

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1367.6∫ 24

0max(0, ~N · ~S) dt = q(24).

As your simulation runs in DIYModeling it can compute q(t). We need to stop thecomputation 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 thedifferential 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 nameyou chose for the variable representing ~S.

Figure 7 shows a screen shot of the Model tab in the Model Editor as you add a newvariable. 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 Ex-pression column.

Figure 8 shows a simulation that should look similar to the one you build now usingthe 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 com-ing from the Sun at the North Pole on the day of the summer solstice in the NorthernHemisphere?

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Figure 7: Adding a New Differential Equation Variable

Figure 8: Daily Energy

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Question 19 What is the total amount of energy in Watt hours per square meter comingfrom 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 comingfrom the Sun at latitude 45◦N on the day of the summer solstice in the Northern Hemi-sphere?

Question 21 What latitude receives the greatest total amount of energy on the day of thesummer solstice in the Northern Hemisphere?

We can take this simulation a step further – computing the annual energy received ata given point on the Earth. Now we want to make the Sun rotate around the Earth in 365days. We’ll use the vector

~S(t) =⟨

cos(

2πt24× 365

), sin

(2πt

24× 365

), 0⟩,

since we’re measuring time in hours, and the integral we need is

1367.6∫ 24×365

0max(0, ~N · ~S) dt.

There is one small problem. The simulation we’ve built thus far is designed for studyingwhat happens over the course of a day. It will run too slowly for simulating what happensover the course of a year. In the Project tab of the Model Editor you can change the speedof 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 in4 seconds in the real world. This is a good compromise. The simulation of a year will takea reasonable amount of time and the Earth won’t be spinning too quickly.

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Figure 9: Changing the Time Scale