Solar geometry and a horizontal sundial




Дата канвертавання27.04.2016
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Physics 103

Solar geometry and a horizontal sundial


It’s another Tuesday morning and you’re sitting through yet another of Nate’s boring labs. The clock on the wall is busted, your cell phone is out of batteries, and the sun shining through the window is making you sleepy. How much longer until 3:50 arrives? You must know…so you whip out your homemade sundial.

In the winter, the days are short and the Sun is low in the horizon of the southern sky. The graphic above shows the Sun's path through the sky on the shortest day of the year, the winter solstice. This is the day when the Sun is the lowest in the southern sky.


During the short winter days the Sun does not rise exactly in the east, but instead rises just south of east and it sets south of west. Each day after the winter solstice, which occurs on December 21st, the Sun's path becomes a little higher in the southern sky. The Sun also begins to rise closer to the east and set closer to the west until we reach the day when it rises exactly east and sets exactly west. This day is called the equinox. In the spring we have the Spring Equinox about March 21st. There is also a Fall Equinox on September 21st.
On the Equinox this is the motion of the Sun through the sky for everyone on earth. Every place on earth experiences a 12 hours day twice a year on the Spring and Fall Equinox. After the Spring Equinox, the Sun still continues to follow a higher and higher path through the sky, with the days growing longer and longer, until it reaches it highest point in the sky on the summer solstice.
On the Summer Solstice, which occurs on June 21, the Sun is at its highest path through the sky and the day is the longest. Because the day is so long the Sun does not rise exactly in the east, but rises to the north of east and sets to the north of west allowing it to be in the sky for a longer period of time.
After the summer solstice the Sun follows a lower and lower path through the sky each day until it reaches the point where it is in the sky for exactly 12 hours again. This is the Fall Equinox. Just like the Spring Equinox, the Sun will rise exactly east and set exactly west on this day and everyone in the world will experience a 12 hour day.
After the Fall Equinox the Sun will continue to follow a lower and lower path through the sky and the days will grow shorter and shorter until it reaches its lowest path and then we are back at the Winter Solstice where we started.

In the diagram displaying the path of the Sun for different times of the year you might have noticed an arrow pointing towards the North Celestial Pole. We need to understand why the location of the North Celestial Pole is important in building our sundials


The North Celestial Pole is the point in the sky about which all the stars seen from the Northern Hemisphere rotate. The North Star, also called Polaris, is located almost exactly at this point in the sky. If you go out at night and find the north star, you will notice that it does not move during the course of the night, while all the other stars do move, they rotate from east to west around the north star.
You might wonder why this is important for building sundials. The Sun is also a star, so the Sun also rotates around the North Celestial Pole (Because we are so close to the Sun, the tilt of the Earth actually varies the exact axis of rotation of the Sun slightly away from the North Celestial Pole.) It is this rotation that will allow us to use the Sun to tell time.
We need to know where the North Celestial Pole is to use our sundials, but during the day, it is too bright to see the north star. How else can we know where to find this special place in the northern sky?
No matter where you live in the Northern Hemisphere there is an easy way to find the north star. As you might have guessed it is located due north, but how high in the sky is it? The north star is up from the horizon exactly an angle equal to your latitude. So if you live at 50 degrees latitude, the north star will be due north, up 50 degrees.
(An average person has about 10 degrees, from little finger to folded thumb, in a closed fist when held an arm's length away.)
The arrow you see pointing toward the North Celestial Pole in the diagram above is close to the axis about which the Sun rotates. All we need to do to make a well-designed horizontal sundial tell time is to point the gnomon (any shadow-casting linear element) of the sundial along this axis. Then, as the Sun rotates around the gnomon it will cast a shadow onto our sundial surface where we will mark the appropriate hourlines. Both the angle of the gnomon and the position of the hourlines depend on the latitude where the sundial will be used.

Sundial Construction Materials:
* A small wood block

* A small nail for a gnomon

* Pencil

* Pen


* Hammer

* Protractor

* A sheet of paper

* A calculator

On a sundial, the gnomon is the piece of the sundial which casts its shadow onto the sundial face. In our project we will use a copper nail as the gnomon.
For a horizontal sundial we want the gnomon to be aligned with the North Celestial Pole so that the sun will rotate around it when we face our sundial due north.
A number of calculations that we must make to create the sundial will require us to use our exact latitude and longitude angles.
The latitude of this room is 4016’15.6” north and the longitude is 7446’48.5”.

Convert these degree,minute, and second angles to decimal angles. There are 60 minutes in 1 degree and 60 seconds in one minute.


Lat.: _________

Long: _________


Preparing the wooden block
To align our gnomon along this axis we must position it such that it makes an angle equal to the latitude angle where it will be used with the horizontal face of the sundial.
The easiest way to get the nail into the wooden block at this angle is to do the following.:
1. Draw a line bisecting the longer side of the rectangular wood block.
2. Use your protractor to draw an angle equal to our latitude from the bottom of the provided index card. Place this card along the bisection and hammer the nail into the wood block about 1 cm from a long edge, making sure that it is parallel to the bisection line and inclined at an angle that matches the latitude angle you’ve drawn.

An accurate sundial?
When solar time became inadequate for railway timekeeping, a system of time zones was initiated, creating 24 time zones, each 15° wide - within which all clocks would tell the same time. (Basic arithmetic: The earth's 360° circumference was divided into 24 one hour segments, and the result was 24 time zones each 15° wide.)
However, the sun moves across each artificial time zone at its own pace.

For your sundial to compensate for our position in this 15° time zone and approach Standard (clock) Time, you would normally need to apply a longitude correction.


This correction is determined by the relationship between the longitude of your sundial and the longitude of the central meridian of our time zone.
What is the central meridian of the Eastern U.S. time zone? _________

(hint, prime meridian at Greenwich, England = 0 degrees)


How many decimal degrees are we from this central meridian? ________
If your longitude is west of your central meridian, add four minutes to the solar time for each degree. If you are east of your central meridian, subtract. In other words, the hour lines on our sundial will read the solar time at our location, not the standard time (which is set as the solar time at the central meridian), unless we adjust their angles slightly. For instance, if we were slightly east of the central meridian, the sun would be just west of due south at the clock-time noon (when the central meridian reaches solar noon).
How many (clock) minutes away from the time at the central meridian are we here in central NJ?

__________


If you did the calculation properly, you will find that because we’re so close to the 75 degree longitude line, our local solar time is less than one minute ahead of the clock time of the eastern time zone. This makes our sundial much easier to make, since the longitude correction is essentially negligible.

Making the hourlines
On a horizontal sundial the position of the lines that indicate what hour it is depend on the latitude. We will calculate these angles and then use a protractor to create a template for placing the lines on the face of our sundial.
The equation below determines at what angle, theta, to draw each hour line. For the angle L you will use your latitude angle, and the angle H is the hour angle, which changes depending on which hour line you are calculating and upon your longitude correction.

The sun travels 360 degrees in 24 hours, or 15 degrees each hour. The hour angle for each given hour line is 15 degrees multiplied by how many lines away from the noon line you are plus the longitude correction in hour-fraction. (For instance, 2 o'clock is two lines away from the noon line, so the hour angle for 2 o'clock is (2 hrs + long. correction)x 15 degrees.)


Use the previous questions and the equation below to complete the following table. Once you have calculated the angle values you are ready to construct your hour line template.


Where H = (Time – Time @noon at the time zone central meridian)x15 degrees



A

B

C

D

Eastern Standard time here

Time change from solar noon [A-12]

(in decimal hrs)



Hour Angle, H
[H=15*C]

(degrees)



Sundial Hour Line angle ()

From gnomon

[use eqn. above]

(degrees)



12 pm (12:00)










1 pm (13:00)










2 pm (14:00)










3 pm (15:00)










4 pm (16:00)










5 pm (17:00)










6 pm (18:00)










11 am










10 am










9 am










8 am










7 am










6 am










To begin construction of the template, place your protractor on a sheet of paper such that the straight edge is near the bottom. Draw a horizontal line on the paper using the bottom edge of the protractor. Without moving the protractor make a pencil tick at the 90 degree mark, which should be at the top of your protractor. Now turn the protractor sideways and use the long edge to draw a vertical line from the tick mark down to the horizontal line. This vertical line represents the solar noon time.





You can now use the values you calculated to draw each hour line at the appropriate angle from the vertical noon line. Begin by placing your protractor sideways on your paper so that it looks like the letter D. You want the long edge along the vertical line and the middle of that long edge where the vertical and horizontal lines cross. Hold the protractor very still and using the numbers written along the curve of the protractor make a tick with your pencil at each of the angle values in the last column of the table above.

Now use the straight edge of your protractor to connect each tick mark to the intersection of the horizontal and vertical lines. You final template should look like the one at the beginning of this section.


Once you have a hard copy of the hour line template, place your wooden block on top of the image such that the center of the block, where the nail hole is, is directly over the intersection of the horizontal and vertical lines and the gnomon is pointing up the noon line.
Holding the block very still make 13 pencil marks on the top face of the wood, each mark inline with the appropriate hourline. It is best to start with pencil so you can redo any mistakes.
Once you have made all of the hour marks go over them in pen and write in the appropriate numbers. Because the face of our sundial is small, it is best to include only a few of the hour numbers.
Except in the case of clouds, you are now ready to use your sundial. It should be pretty accurate, but if you want to achieve the best accuracy possible, you must also add a small effect related to the time of year..
Even after the longitude correction is applied, your dial will not consistently tell Standard Time due to The Equation of Time which varies throughout the year. See below.


Finally, step outside, point your gnomon due north, read the time on your sundial and subtract the correction reading from the equation of time chart above. How accurate was your construction? If you do know the time, how could you use your sundial as a navigational tool?


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