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Why is focal length half the radius of
curvature?
It's easier to see graphically if you consider points equidistant from
plane waves. For a sphere, the center of the sphere is trivially the only
point that reflects off of all surfaces and back to the center in the
same distance. For a plane wave hitting a sphere (object at infinite distance),
there is no exact focal point from a spherical surface (that's spherical
aberration) but if you draw a sphere, you can convince yourself that at
least for small angles, the half radius distance is the point that shares
roughly the same reflected distance for a plane wave off a spherical surface.
I think you can see this mathematically if you do an expansion of the
equation for a circle, and leave out the higher order terms (in which
case you end up with a parabola). Answer from Dan Lester.
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How do you line up a telescope with the
sun without damaging your eyes?
The key here is NOT TO LOOK THROUGH THE EYEPIECE! The mirrors concentrate
the light from the sun and can burn your eyes so do not look directly
into the telescope. Cover the front opening (aperture) of the telescope
with a sheet of paper and poke holes in the paper with a pen. This filters
out some of the light so the image will not be too bright. Point the telescope
in the direction of the sun. If you are pointing close to the sun but
not right on it you will see a bright dot on the paper covering the aperture.
This is the image of the sun from the primary mirror. All you have to
do is move the telescope until the spot disappears in the center. This
means it is hitting the secondary mirror, and you should see light projected
from the eyepiece. Hold a paper opposite the eyepiece to view the projected
image. You may have to move the paper closer and farther from the eyepiece
to "focus."
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Why do you have to slide the eyepiece
away from the telescope when you look at the sun?
For those of you projecting the images, namely the sun, you have to make
sure to slide the eyepiece farther away from the telescope than you would
if you were looking at something very distant with your eye. The reason
for this is that when you're in focus for your eye, the rays all come
out parallel. That's great for your eye, because your lens can easily
focus those rays (it's like looking at something far away). However, if
you project these parallel rays onto a piece of paper, then you don't
have another lens, and the rays all get mushed together, giving you one
big blu. If you pull the eyepiece a little farther out, you can make a
big image of the sun come to focus on a piece of paper far away. The more
you pull out the eyepiece, the closer you hold the paper to get the sun
in focus and the smaller the image of the sun is. If you pull it out an
amount equal to the focal length of the eyepiece, (in our case, about
3 cm), then the magnification is 1:1, which means the sun appears the
same size as if you didn't have an eyepiece at all, which is pretty tiny
(about 1 cm). If you pull it out even further, then it'll get even smaller.
So we have about a 3 cm range to play with, which is actually a decent
range to slide the eyepiece around.
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How do I focus a projected image?
Slide the brass tube with the eyepiece in it all the way in (just make
sure it does not go through the hole and crash into the primary mirror!).
Then pull it out around 2 cm. Put the paper onto which you are projecting
the image behind the eyepiece and move it back and forth until you focus
the sun. If you can't get it to focus, pull out the eyepiece a little
more until you can. Then it's just a Goldilocks problem. If the sun is
too small, push they eyepiece in a bit, move the pad to refocus the projected
image, repeat until you have the size you want. If the projected image
is too big, pull the eyepice out a bit, move the pad to refocus, repeat
as necessary.
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How do I time the sun?
The easiest way I found to do this, which works rather nicely, is to put
the sun somewhere in the field of view, away from any edge. Then just
let it set for a minute or so until you see one edge starting to fade
a little. Start timing. Watch it until the sun completely disappears from
the field of view. Stop timing. You've just measured how long it takes
the sun to move an angle equal to its own apparent (angular) diameter.
I got values between 2:00 minutes and 2:20, mostly closer to 2:20. The
time obviously depends on when you decide the sun is first reaching the
edge of the field and when you decide it's totally gone. That's totally
fine and is an important lesson in and of itself. The idea of uncertainty
and measurement error, random and systematic (both are at work in this
measurement) is one of the central ideas we try to get across in Research
Methods class. With these data, you have ~ 2:20 minutes/(length of day)
= (angle subtended by the sun)/(angle of entire circle), a nice set of
proportions with which to work and manipulate however you want. NOTE:
I did this with the eyepiece, which works better than without because
1) you can keep the paper farther away from the telescope, making it easier
to see and 2) you can project a bigger image of the sun, making it easier
and more accurate to time when it crosses the edge of the field of view.
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How do I measure the angle of the sun
relative to the field of view?
Once you've found a nice size for the in-focus sun, just trace out the
disk on the sheet of paper. Then move to another part of the same sheet
of paper but MAKE SURE YOU KEEP THE PAPER THE SAME DISTANCE FROM THE EYEPIECE.
Move the telescope around so the the sun appears to be on the edge of
the field of view, maybe half off it. You can trace out a small arc of
the large circle which is the field of view. Move the telescope a little
so the sun is still ~half off the field but adjacent to the last position.
Trace out another small arc. By doing this around the whole field of view,
you can trace out the entire field of view in these pieces. Measure the
diameter of the sun, measure the diameter of the field of view, and the
ratio is the fraction of the field of view taken up by the sun. If you
know the angle of the field of view, you can then calculate the angle
of the sun. with the angle of the sun, knowing either the size or distance,
you can get the other.
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How do I figure out the diameter of
the field of view?
Have a student hold up a meter stick at a known distance from the telescope
(30 meters works well). Have the other student look through the telescope
and determine how much of the meter stick fills the field of view. At
30 meters, about 45 cm fills the field of view. At 50-60 meters, the meter
stick just fills the field of view. You have a simple size/distance ratio:
angle=size/distance.
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