Finding the Diameter of the Sun 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])

First off, have you read HOW TO DO ACTIVITIES,  HOW TO WRITE UP ACTIVITIES (5-Point Form) and HOW TO SUBMIT ACTIVITIES? If not, read these first or you may lose points. (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.(To see the set up another student used, click here: student example)

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 in which a small hole has been punched.  (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.)

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 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 will use the form in the online submission file.)

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.

END

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Copyright 2006 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 any profit-making activity without express written consent.

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