Earth-Sun Relationships/Navigation

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The Sun Lab is used in our Natural Systems courses and Tropical Marine Ecology.

Compiled by Hays Cummins, Western Program, Miami University

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It is 1:04:01 PM on Wednesday, September 28, 2016. This page has served 135812 sun worshippers and was last updated on Wednesday, May 7, 2014.


Lecture Outline

Introduction

The sun provides nearly all the energy that powers atmospheric and oceanic circulations, that generates the geologic processes of weathering, erosion, transportation of materials and deposition, and that supports and sustains life and growth within the biosphere (Except for deep-sea vents communities.)

The sun makes "the Weather World Go Round."

The earth, with its elliptical orbit around the sun, rotates on an inclined axis and receives the sun's energy in a rhythmic pattern through time. Because of the inclination of the earth's axis of rotation and because the earth's axis at any point in the earth's orbit around the sun is parallel to the axis at any such point, the most direct rays of the sun are shifted from north to south during the year (the declination of the sun). This declination brings about the seasons with their different intensities of solar radiation and varying lengths of daylight and darkness.

See a Fantastic Movie of Seasonal Change as Seen from Space OR Why an axial tilt of 23.5 is important!

Sun Angles

The angle at which the sun's rays strike the earth's surface is a major factor in the amount of energy received per unit of surface area. More direct rays provide more concentrated energy. Sun angles are also useful in navigation for determining latitudinal position.

Position of the Sun

The position of the sun during the course of a day can be studied by using the shadow cast by a vertical stick. The line formed by successive shadow points can be analyzed to yield the latitude at the observing site. The sun differs from the stars by having a variable declination during the course of one earth year, varying through the year from about +23.5 degrees to -23.5 degrees. The important thing to understand is that on any given day the declination of the sun is known and that declination changes on a daily basis.

A solar prominence is a cloud of solar gas held above the Sun's surface by the Sun's magnetic field. The Earth would easily fit under one of the loops of the prominence shown in the above picture. Source: Astronomy Picture of the Day

Problems at Sea/Navigation

The most critical requirement of all marine work may be precise positioning, because data reported from generalized positions are virtually useless to others wishing to follow up on previous work. Unfortunately, in the absence of landmarks, there are no fixed points of reference on the ocean surface. The coordinates of latitude and longitude are essential in navigation. The lines of latitude are also called parallels of latitude, because they are all parallel to the equator and to each other. Measured in degrees of arc along a circle, they specify the angular distance north or south of the equator, from 0 degrees at the equator to 90 degrees at either pole. Each degree is divided into 60 minutes of arc (1 degree = 60'; 1 minute = 1 nautical mile or 1.15 statute miles) and each minute into 60 seconds ( 1' = 60"). Latitude is recorded with its hemisphere notation, north (90 N) or south(45 S); lines of longitude, or the meridians, are also expressed in degrees and refer to the angular distance on the earth measured from the prime meridian (0 degrees) at Greenwich, England, east or west through 180 degrees. Another term frequently used in navigation is the great circle. It refers to any circle traced on the surface of a sphere by a plane that passes through the center of that sphere. All longitudinal lines are great circles. Small circles refer to the line of intersection of a sphere and a plane that does not pass through the center of the sphere. All lines of latitude, with the exception of the equator, are small circles.

Finding the Latitude of Your Location Using a Meter Stick

To determine a ship's latitudinal location on the ocean surface, sailors turned to celestial objects(the distant stars and the sun) to determine location. For navigational purposes, it is possible to determine the latitudinal location from sun angle relationships. Usually a sextant is used to determine the altitude angle of the sun or other celestial object (the angle of the object above the horizon). We will use a meter stick and the resulting sun's shadow along with some simple trigonometry to determine the sun's altitude and zenith angles and the latitude of our location.

How can the sun be used to determine latitude? By observing the altitude angle and zenith angle of the local noon sun, one can determine the latitudinal position of the observer. Remember, the sun's declination is a known quantity; we know where on the earth's surface the sun is directly overhead on every day of the year. We will obtain the sun's declination for the date of this experiment, using the computer program,Voyager.

The cartoon below illustrates the differences between the altitude angle and zenith angle. A meter stick or any other straight object (not to scale!) is used to cast a shadow on the earth's surface. Pay particular attention to the significance of the zenith angle. It is of prime importance in the determination of one's latitude. The zenith angle is the # degrees of latitude you are from the location where the sun is directlly overhead at your local noon. The altitude angle is the number of degrees the sun is above the horizon. The sum of the altitude angle and zenith angle is equal to 90 degrees.


How do I determine what the altitude angle of the sun is with only a meter stick? As the sun's rays strike the meter stick, a shadow is cast. We are interested in measuring the shadow when the sun is at it's highest point in the sky. This event takes place at local Noon. If we didn't have a watch, we would know when the sun crossed our meridian because the shadow would be at its shortest length and be pointing (in which direction?). Since we know the length of the meter stick (100 cm), we then determine the length of the shadow in centimeters. To determine the altitude angle of the sun above the horizon, one must then use simple trigonometry.

Just Where is the Sun Directly Overhead?

Two apparent motions: the earth's rotation around the sun during the course of a year and the daily rotation of the earth. Figures from NASA's From Stargazers to Starships and Honolulu Community College's The Nature of Physical Science

So, where is the sun directly overhead at local noon on any given day? There are a couple of ways to approach this. One is to estimate where the sun is located by calculating its average daily movement during the course of one earth year. Now, you might be thinking, "What the heck is he talking about?" But, there is a method to my madness! Would you agree, in a general way, that the sun "travels" from the Tropic of Cancer (23.5 N) to the Tropic of Capricorn (23.5 S) and back to the Tropic of Cancer in one earth year (365 days)? If this is true, then the sun travels 47 degrees of latitude * 2 (to compensate for the round trip!)=94 degrees of latitude. It is a simple calculation to compute the average daily movement of the sun (94 degrees/365 days = ~.258 degrees/day or ~15.5 nautical miles/day.

What we need is a frame of reference! Something we can sink our teeth into. Hmmmmmm. Using the above assumptions, and the fact that we know the exact locations of the sun in the northern hemisphere during the spring equinox (local noon-sun over the equator 0 degrees), summer solstice (local noon-sun over the Tropic of Cancer 23.5 N), fall equinox (local noon-sun over the equator 0 degrees), and winter solstice (local noon-sun over the Tropic of Capricorn 23.5 S), you should be able to estimate where the sun is directly overhead at local noon on any given day of the year. These calculations are extremely important when determining your location's latitude!

But, having said all of the above, it's time to admit that I'm not exactly correct in my reasoning. "How can that be so? You're the professor and you're supposed to know everything!" you might shout out. Life is hard--deal with it.......

Enough of that, though. The problem is that the earth has an elliptical orbit around the sun--sometimes the earth is closer (our northern hemisphere winter) while at other times it's further away (northern hemipshere summer). This causes the earth to move faster when it's closer to the sun and slower when it's further away. Johannes Kepler was the first to mathematically define the Laws of Planetary Motion--the one we've been talking about is Keplers 2nd Law. So, the above calculations will probably be incorrect! One of your tasks is to determine how incorrect your calculated sun's Declination (latitude where the sun is directly overhead at local noon) is when compared with the actual declination of the sun. How close will it be?

You can get the sun's actual daily position from the US NAVAL OBSERVATORY. They've got tons of stuff here and I encourage you to explore! But, if you'd like to home-in on the sun's daily position at local noon, go to Data for Solar System Bodies and Bright Stars. Click on the Web version of MICA. And, there you go! Look for the sun's declination. That is the latitude where the sun will be directly overhead at local noon!

You can visualize the movement of the sun in the course of a year by examining what is called the Analemma. The image below is the Analemma for Washington, D.C. I obtained this graph from the US Naval Observatory, a really, really cool site!

The sun moves faster in the winter months and slower in the summer months (northern hemisphere). Click on the image for a larger view!

Now that you have the "Scoop," can you determine your latitude?:

1) Divide into teams containing 4 people.

2) Examine the globes in the lab. What is the significance of the Antarctic, Arctic, Tropics of Cancer and Capricorn small circles that encompass the globe? At what latitudes are they found? Why?

3) Determine the altitude angle of the sun at our location. Give yourself about 90 minutes to do this experiment. Record the times and lengths of shadows measured. Measure at least 8 different shadow lengths. At what time did local noon occur? Plot the relationship between shadow length and minutes before and after local noon.

4) Does the shortest noon shadow length point due north ? Use a compass to find out.

5)Over what latitude is the sun directly overhead at their local noon on the day you are taking these measurements?

6)Use the information above and below to determine the latitude of your location!! You can do it!

Pythagorean Theorem


Use the pythagorean theorem to help you determine the altitude angle of the sun. The pythagorean theorem applies to a right triangle (that is, any triangle having a right angle). The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, or legs. In equation form,

C2 =A2 + B2

Trigonometric Ratios

Let A be some angle in standard position (see figure below). From any point R on the terminal side of the angle we draw a perpendicular to the x axis. This forms a right triangle ORS. Side OR is the hypotenuse of the triangle, and has a length p; RS is the side opposite angle A and has a length y; OS is adjacent to angle A and has a length x. The six trigonometric ratios are now defined in terms of the coordinates of point R, as well as in terms of the right triangle ORS.

Examine the figure below,


Which of the following trig ratios will you use to determine the altitude angle of the sun?

Sine A = Sin A= Y/P = F(opposite side,hypotenuse)

Cosine A= cos = X/P =F(side adjacent,hypotenuse)

Tangent A = tan A = Y/X = F(opposite side,adjacent)

Cotangent A= cot A = x/y = F(adjacent side,opposite side)

Secant A = sec A = P/X = F(hypotenuse,adjacent)

Cosecant A = csc A= p/y = F(hypotenuse,opposite side)

Another way to visualize the model and the same trig relationships!!

A Lesson in Human Creativity: Finding your Longitude


Even though we now know our latitude, if we were on a ship without any instrumentation or known landmarks, we wouldn't have any idea as to our longitude.There are 15 degrees of longitude for each hour of the day since the earth rotates through 360 every 24 hours ( 360/24 hours = 15/hour). Longitude is determined by measuring the difference in time that noon occurs at any given location and the standard reference longitude of 0 degrees at Greenwich, England. In the 18th century, Captain James Cook was one of the first navigators to sail the oceans with the primary purpose of learning their natural history. He also made the first accurate maps using the chronometer developed by John Harrison. This time piece shows the time on the Greenwich Meridian - the prime meridian at 0 degrees longitude. Since the earth rotates through 15 degrees of longitude per hour, a ship's captain could easily determine his longitude each day at local noon (when the sun crosses the meridian directly overhead) by measuring the difference in time between Greenwich and his local noon. A three hour difference in time between his location and Greenwich can result in what longitudes?


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US Naval Observatory- Astronomical Applications computes, from fundamental astronomical reference data, the position, brightness, and other observable characteristics of celestial bodies, as well as the circumstances of astronomical phenomena.

Questions:

1. Now that you know our latitude to the nearest degree, what would the altitude angle of the sun be at local noon on Dec.22 (Winter Solstice), June 21 (Summer Solstice), March 21 (Vernal equinox), and Sep. 22/23 (Fall equinox) in Oxford ? At the Galapagos Islands?

2. How does the apparent movement of the sun during the course of a year effect the heat balance of the earth?

3. What is the significance of the Tropics of Cancer & Capricorn as well as the Arctic & Antarctic circles? Why are they found on the globe in these particular locations ?

4. If a ship's captain observes a Greenwich meridian time of 1530 hours on a chronometer at local noon, what is the ship's longitude? What would the longitude be if the chronometer read 900 hours at local noon?

5. Assume that your location is 10 N latitude. At 12:00 noon, the sun crosses your meridian and you notice that the zenith angle of the sun is 10 degrees (the altitude angle of the sun must be ? ). Your best friend lives on the equator directly south of you; he or she notices that on the same day and time the sun casts no shadow when it crosses directly overhead. If the earth was flat would this be the case? Why?

6. Based on your current knowledge, can you determine the circumference of the earth?


Our Sun--Quicktime Movies of Eclipses and Solar Activity

A fantastic GOES 10 satellite loop (animated gif) of the Feb. 26, 1998 Solar Eclipse! (From GOES Hot Stuff.)

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