The Diameter of the Sun |
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- Be sure you have read the file on how to Do activities
- Be sure you have read the file on how to Write-up activities
- Be sure you have read the file on how to Submit activities
- Start early enough so that you can finish on time. There are no time extensions
- Ask if you don't understand something
- Check your work when possible (it usually is), and ask yourself, "Does this seem reasonable?"
- You are allowed and even encouraged to work with a partner, but each of you should take your own measurements and prepare your own individual report to turn in (no copies).
The image at left is the most recent solar image from the SOHO solar satellite. If the image is missing or says "CCD Bakeout," it may be due to a temporary problem with the satellite. (SOHO Michelson Doppler Imager (MDI) 6767 Å continuum images from Stanford University])
(You must allow plenty of time to do this activity. It's not that the activity itself takes all that long, but you must have a clear sunny day. Please read this entire file before you start. If you do not understand anything, ask.)
SYNOPSIS
Reflect sunlight from a small mirror onto a flat, light-colored wall or
other
flat surface as directed. Measure the distance from the mirror to the
image on
the wall; and measure the diameter of the image. Plug the two
measurements into
a simple formula to calculate the diameter of the Sun.
BACKGROUND
The Sun's distance – and hence its diameter – was not known accurately
until
the famous British naval Captain James Cook and others observed the
1769 transit
of Venus. With views from various locations, astronomers could
determine the
parallax and hence the absolute distance to Venus. Knowing the relative
distance
to the planets from Kepler's Laws, they could then work out the
distance to the
Sun. There was some discrepancy in the results from various observers,
but a
reasonably accurate distance to the Sun (about 150 million kilometers)
was
derived from the data in 1835 by the astronomer Encke. Here the radius
of the
orbit, which we now know averages about 150 million kilometers, is
designated by
the letter "L." Knowing the radius of the orbit and by measuring the
apparent size of the sun in the sky, astronomers could easily determine
the
diameter of the sun using trigonometry or the simple geometry of this
activity.
NOTE:It is perfectly safe to look at the image on the wall, but do not look directly at the Sun or into the reflection from the mirror. The diagram above is just one possibility -- you don't have to do it exactly like this. You can place the mirror just outside the window or even on a windowsill where the Sun is shining. Or you can stand in a driveway and reflect it into a garage, or just set it up outside and reflect onto the wall of a building in the shade.
In the diagram above, the distance from the sun to the mirror ( L ) bears the same relationship to the sun's diameter ( D ) as the distance from the mirror to the solar image ( l ) does to the diameter of the solar image ( d ). You know that L averages 150 million km. To find D , the diameter of the sun, you only need to set up a mirror reflection as indicated, and then measure l and d .
D/L=d/l
or
D=(d/l)*L
The slash (/) represents division, whereas the asterisk (*) means "multiplied by."
PROCEDURE
A flat compact mirror will work fine, but to keep the distance "l"
down to a workable figure, you must stop the mirror down to a smaller
size,
perhaps 6-8 mm, by covering the mirror with paper or cardboard in which
a small hole
has been
punched. The paper or cardboard should be thin and dark to avoid any
glare. Black or dark construction paper is perfect. If you use
cardboard, don't use the corrogated kind from a box. Use only thin
cardboard, such as what is sometimes called "shirt" cardboard. (Experiment
with sizes and distances to
find something that works for you. Also keep in mind that the ratio of
the size
of the reflection distance ("d") to the size of the mirror cutout
should be in the order of 800 or 1000 to one. A 6 meter reflection from
a 7 mm
mirror fits this ratio. Do NOT use reflection
distances ("l")
of less than 5 meters or more than 7 meters. Be sure that
whatever mirror
you use is flat and does not magnify.)
The fact that we are using a cut-out hole in front of the mirror confuses some students, and some apparently decide that it is not important and follow this step. Rather, they just use the whole mirror. They are wrong, as the cut-out is an absolutely essential step. You will lose points if you fail to follow the instructions. Setting up the cutout and mirror forms a type of simple lens utilizing the same principles as a pinhole camera, except that we are using a reflection rather than a projection through the hole. If you want to know more, please ask. And please never assume that something is not important just because you don't understand it.
You must figure out a way to hold the mirror steady so you or a friend can go measure the diameter of the reflected image. Make the hole in the paper or cardboard (to show the mirror) 6 to 8 millimeters across, and then arrange the distance "l" so that it is about 5-7 meters. This is for practicality. The exact size and shape of the hole in the cardboard is not important, but it should be roughly 6-8 mm across, and the reflection distance ("l") should be 5 - 7 meters. The larger the size of the mirror (or cutout), the longer the reflection distance must be.
(Save yourself some trouble by reading the Example and Notes below before you start any measurements!)
You should reflect the light onto a light-colored surface, but one that is in an darkened area. This is so the image will show up better. Do not use a dark-colored surface or wall because it will make it hard to see. Use common sense.
With a mirror (cutout) of about 6-8 millimeters, a reflection throw ( l ) of about 6 meters should work. Keep in mind that the reflected image of the Sun will be faint, so it needs to be reflected into a darkened area. If your distance l is too short, the image will be brighter but the edges will be fuzzy. If l is too great, the image will be too faint to measure. If your cutout is too small, the image will be too faint. If it is too large, the edges of the reflected image will be fuzzy. Experiment to find an arrangement where the image is large enough, sharp enough, and bright enough to measure accurately.
Make at least 3 measurements both of the distance l and the diameter d . (Just measure the distance and diameter 3 times, trying to be as accurate as possible. You likely will come up with slightly different values each time.) Put the data in a table, and apply the formula above to determine the diameter of the sun. Then average your results. Be sure to take the average of your results -- do not average the measurements you take. Treat this as a regular lab activity, supplying all five components as usual. Consider and describe possible sources of error.
HOW TO
MEASURE: It is difficult to
measure the size of the reflection on a wall using a ruler. Instead,
print out
these two pages of circles: 
If you follow the directions, then your reflection on the wall should closely fit one of the circles. (If it is just a bit larger than one circle and a bit smaller than another, you can interpolate). Find the closest match and then measure the diameter of the circle as your "d." Use a metric ruler with millimeters (mm). Measure "l" in meters and convert to mm (1 meter = 1000 mm) If the reflection on the wall is oval (it should be round) or is significantly larger or smaller than any of the circles provided, then your set up is wrong and needs to be fixed.
Check your results against your textbook(s), some other reference, or through a simple online search. If you fail to check your results (when it is easily possible), you may lose points. I am not looking for extreme precision, but if your result is not within about 10 percent of the accepted value, then you did something wrong. If you don't understand something, ask.
Here is a table to put your data and results in. Remember, do NOT average your data. Instead, figure out "D" for each set of measurements, and then average your results for "D." That will go in the green box and is your answer. (NOTE: The table example provided here is for lecture-class students. Online students do not have to use this, but instead online submission form. For suggestions on a submitting data not in a table, see non-table data.)
| "d" (in mm) | "l" (in mm) | D=(d/l)*150,000,000 km | |
| 1 | |||
| 2 | |||
| 3 | |||
| Average: | (don't average data) | (don't average data) |
Remember that your "d" and "l" must be in the same units (e.g., millimeters or "mm")
By the way, you know that the distance to the sun varies through the year, ranging from about 146 million kilometers in early January, to about 151million kilometers in July. You could find out the exact distance on the day of your observation, but the level of accuracy of this observation doesn't warrant it. It is sufficient to use the figure of 150 million kilometers.
Recap:
- Find a non-magnifying mirror, some cardboard, a ruler and tape measure (preferably metric)
- Using the cardboard, stop down the mirror as directed
- Find a location where you can properly aim the reflection from the mirror on a flat wall
- Stand 5 to 7 meters from the wall and measure the distance carefully
- Project the reflected image of the Sun onto the wall and measure its diameter
- Take at least 3 sets of measurements, and average the results
- Check your result against the textbook or other reliable source
- Write up your report according to the Write-up instructions.
- Submit your report before the deadline, according to the instructions provided.
END
| EXAMPLE:
Let's say that you make your
measurements from 6 meters (6000 mm) and that one of your
measurements is 64 mm. In this case, the "d" is 64 mm and the "l" is
6000 mm. The ratio (or equation!) becomes:
D=(64mm/6000mm)*L. D=(64mm/6000mm)*150,000,000 km. D=(64/6000)*150,000,000 km. Now, 64 divided by 6000 equals 0.01067 (rounded). Some people mess themselves up by doing it the wrong way. Instead of dividing 64 by 6000, they divide 6000 by 64. NO! Read it the right way. On a calculator, you would enter "64/6000" like this: 64 ÷ 6000 = answer. Punch in "64" then the divide
symbol (÷ or /), then "6000" followed by "=" D = 0.01067 * 150 million km. Just multiply (*) the 150 million by 0.01067. On a calculator, punch in "150000000" (no commas) followed by the multiply symbol (usually "X" or "*" on a calculator) followed by "=" to get: 1,600,500 (rounded a bit). Putting in the commas you get: 1,600,500 km. Your answer would be 1,600,500 km. Then repeat this 2 more times, average your results for the final answer. Be sure to show all pertinent data and your calculations. NOTE: This is an incorrect answer because the data I used (64 mm and 6000) were incorrect. I just used it for the example. Follow this procedure with your own measurements and you should have no trouble. Always watch the units (meters, millimeters, etc.) you are working in. Don't confuse them and don't try to do something like divide millimeters by meters or multiply kilometers by meters. If you have a ratio, such as "A/B", then be sure that both A and B are in the same units (such as inches), and then the units will cancel out in the ratio to leave you a "pure" dimensionless number like 0.25 or 245 without any inches. Also note that there is no need whatsoever to change everything into miles as some students do. For instance, you do NOT need to change a measurement like 64 millimeters into miles. It just complicates things and makes it much more prone to mistakes. And please don't confuse miles with kilometers. Your book usually uses kilometers (km) instead of miles. A mile is longer than a kilometer. It takes 1.609 kilometers to make 1 mile. If you get them confused your answer will be way off, so stay with millimeters, meters and kilometers. Don't make life any harder than it is. PRACTICAL NOTE : It will take a little practice to learn how to aim the mirror where you want it, and again, it is very convenient to have someone help you with this. I suggest that you try this handheld first to get a feel for it. After that find a way to aim the mirror and have it stay in place at least for a minute or two while you make your measurements. No one can hold it stead enough by hand to do this. It needs to be mounted firmly in some manner. I mounted the mirror on an old camera and mounted that on a tripod for easy aiming. You could try devising a mounting with a cardboard box, or by holding the mirror with a large lump of modeling clay. Be inventive.The important points are that the mirror must be small compared to the distance of reflection ("l"), which should be about 6 meters, and that you must get an image bright enough to measure. The exact size of the mirror is not important as that measurement does not figure in. It is the ratio of the distance of reflection ("l") to the size of the reflected image ("d") that is important. The actual size of the mirror determines how easy or hard the image will be to see, but if your reflection distance is large enough, the size of the mirror does not affect the ratio. The projected (reflected) image should be round. If it is not round, your measurements will be wrong. An oblong image indicates a problem with your set up. If you need to, adjust your set up until you get a round image. MATH NOTE : You should take your measurements in metric units -- that is, meters and millimeters. If you do not understand how to do this, check the measurement help page or some other reference: measurements. (There is also lots of information here: http://lamar.colostate.edu/~hillger/ )If you can't find a metric ruler or your tape measure does not have metric units, you will have to convert into metric units. One inch is 25.4 millimeters, and one foot is about 305 millimeters. In any event, your "d" and "l" must be in the same units. In this and all activities and in fact anything related to this class, if you measure in the English system (feet, inches) and then convert into the Metric system (meters, millimeters), you MUST include your original measurements (English) along withe the converted measurements in your write up! |
Copyright 2009 by Final Copy, Inc. All rights reserved.
This activity may be
reproduced for classroom use, but may not be republished in any form,
or used in
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Final Copy, Inc.