ESCI 110: Earth-Sun Relationships
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Exercise 4
Earth-Sun Relationships and Determining Latitude
Introduction
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:
ƒ Describe the conditions characteristic of each of the solstices and equinoxes and be able
to-describe and explain the changing seasons.
ƒ Explain the reversal of seasons that exists between the northern and southern hemispheres.
ƒ Be able to explain the ever changing duration of daylight and why some areas experience
24 hours of daylight or darkness.
ƒ Determine the latitude of a point knowing the date and the angle of the sun above the
horizon.
ƒ Find the latitude of a point by observation of the north star.
Materials
ƒ
ƒ
ƒ
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.
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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: ___________________
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Background Information
Fig. 4-1. 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.
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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.
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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:
A&B
B&C
C&D
D&A
N. Hemisphere
S. Hemisphere
__________;
__________;
__________;
__________;
__________;
__________;
__________;
__________;
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
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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 = 90O – 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?
__________
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Figure 4-4. The analema.
Use the analema to determine the sun position for the date on which you measured the
sun’s altitude:
________________
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Fig. 4-5. 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
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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 20O 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} + 23.5 O S {determined from the analema}
Zenith angle = 43.5 O
Draw out
on board
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?
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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?
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f. It is April 1. An observer is located at 40° S. What is the altitude of the noon sun?
Fig. 4-6. Locating your latitude if you and the sun are located in the same hemisphere.
How do we
know?
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:
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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 = 90° - 30°
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 + 23.5°N
LP = 60° - 23.5°
LP = 36.5° S
Or
60° = LP – 23.5° N
LP = 60° + 23.5°
LP = 83.5° N
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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?
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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.
__________
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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