 ## 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 figure-eight? 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.) ## Part One:

### 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 right-hand 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 12-13 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 non-circularity 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 Earth-Sun 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 arc-seconds in a radian).```
```	Given:  The actual physical diameter of the Sun is
diam = 1.4 x 109 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 semi-major axis length a of the Earth's orbit.
```	Given:  The "perihelion" distance is RMIN=a(1-e)
and the "aphelion" distance is RMAX=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, RAVG = (RMAX + RMIN) / 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.