Measuring the Distance to the Sun
A printable version of this lesson can be found
here.
Discussion:
What is the shape of the Earth's path around the Sun? Is it a
circle? Is it an ellipse? A figureeight? How do you know?
If the Earth's path were circular, how could you tell? If the Earth's path
were elliptical, how could you tell? Would the appearance of the Sun
give you a clue? Let's look at some pictures of the Sun and see if
there's anything in the data that helps us. (The students may guess that
the Earth's path is circular, or they may have heard somewhere that the
path is elliptical. The instructor should guide the students, if they don't
come up with the hypothesis themselves, to the idea that a circular path
means a constant distance to the Sun and hence a constant apparent solar
diameter. It is this hypothesis that will be tested.)
Hypothesis:
You know that the Earth goes around
the Sun, once per year. Let's suppose that the path the Earth takes,
its "orbit", is a circle. If the Earth's orbit is perfectly
circular, then that means the Earth is always the
same distance from the Sun. And if it's always the same distance
away, this means that the Sun would always
appear to have the same size in the sky. Another way to say this is
that the Sun's angular diameter would
appear to be constant.
Investigation:
How can we test this hypothesis? Start by grabbing four
pictures of the Sun, taken a few months
apart. (Ask yourself: Why four? Why not just any two, six
months apart?) Load the four pictures into your browser, or your image
processor (NIH Image, ImagePC,
etc.), and LOOK. Is the Sun the same size in all four images?
There are a couple of ways to compare the relative sizes. Most
straightforward is to make a ruler out of a piece of paper and hold it up
to the computer screen; use the same ruler to measure all four images.
If you have an image processor like NIH Image or ImagePC,
another way is to put the cursor on the left and then righthand
edges of the Sun, and then use the printout of the
cursor position to count how many pixels across is the Sun.
More Technical: Also in an image processor like
NIH Image or ImagePC, you can use the "Image Math" selection
under the "Process" menu. Choose
one image (say, January) as the 'standard' and then take turns subtracting
the other images away from the 'standard.' What do you see?
Discussion:
If the Sun is the same diameter in every image, then
the comparison of the four images should reveal only where
sunspots have appeared/disappeared, but nothing significant about the Sun's
edge. But if the solar diameter has changed in the months between when
the images were taken, you should be able to detect this with your paper
ruler. If you try the More Technical exercise, subtracting one
image from another, then the result of the subtraction should show a
ring, an annulus, that corresponds to the change in angular
diameter. An example of this subtraction is provided at URL:
http://solar.physics.montana.edu/YPOP/Classroom/Lessons/Eccentricity/Images/subtraction.gif
In this example, the 'July' image has been subtracted from the 'January'
image. By zooming in on this ring (use the "magnifying glass" cursor option
of your image processor), you can
measure the width of the annulus  simply count how many pixels make up
the thickness of the ring.
Clearly, the Sun does not appear to have the same diameter all
year long. (You ought to see a variation of 1213 pixels in diameter over
the course of a year, assuming that you're using 512x512 images. More
than 18 pixels variation is suspect.) This means that either (a) the
Earth's orbit is perfectly circular and the Sun is actually changing in
size (not very likely), or (b) that the Earth's orbit is not perfectly
circular (more likely  it's a fact).
Query:
When does the Sun look the biggest: January or July? What
does that say about the relationship between the closeness of the Earth to
the Sun and the timing of the seasons in the Earth's Northern Hemisphere?
Part Two:
Background:
The path that the Earth (and the other planets too, for that
matter) takes around the Sun is not a perfect circle, but an ellipse. The
amount of noncircularity is called the "eccentricity;" a more eccentric
ellipse is more stretched out, less circular. In the above exercise,
you demonstrated that the Earth's orbit is not circular  sometimes we
are closer to the Sun than at other times. The next question is, "How
much closer?"
Exercise:
Pick two images of the Sun; use the one that has the
largest apparent solar diameter and the one that has the smallest. These
should be about six months apart. (Why?) Use these two images
to find the variation in the EarthSun distance R. Given: In a 512x512 image from Yohkoh,
each pixel corresponds to 4.9 seconds of arc
(a second is 1/3600 of a degree  there are
206265 arcseconds in a radian).
Given: The actual physical diameter of the Sun is
diam = 1.4 x 10^{9} meters
(1.4 million kilometers).
You will find it useful to know that for small angles ANG, the
relationship between diam, ANG, and R is
diam = R x ANG as long as the angle
ANG is measured in radians.
If you have an image processor like NIH Image or ImagePC,
you can use your computer's cursor to measure the Sun's diameter in
pixels. If you don't have one of those software packages, use these
numbers: in the January 1992 image, the Sun's angular diameter is 396
pixels; in the July 1992 image, the Sun's angular diameter is 383 pixels.
Part Three:
Exercise:
Calculate the eccentricity e and
semimajor axis length a of the
Earth's orbit. Given: The "perihelion" distance is R_{MIN}=a(1e)
and the "aphelion" distance is R_{MAX}=a(1+e)
Query:
The eccentricity of the Earth's orbit is pretty small.
Mercury's eccentricity is more than 12 times bigger.
How much variation would you expect to see in the Sun's
apparent diameter if you were standing on Mercury?
(For Mercury, R_{AVG} =
(R_{MAX} + R_{MIN}) / 2 = 57.9 million kilometers.)
Lesson designed by the
YPOP Team
For questions about this lesson, please contact
David McKenzie

Selected by the sciLINKS
program, a service of National Science Teachers Association.
Copyright 2001.
