Overview of Activity:
The amount of detail we see in an image of smaller angular size with some set physical dimensions (a "close up") is limited by the quality of the original image, determined as the angular size of each pixel. This size is due in turn to the quality of the telescope and image-recording devices used. For the smallest images, pixels must be duplicated in a set pattern around each original pixel to provide enough pixels to fill the 300 x 300 image frame. This can make the image look "blocky." For very large images, pixels may have to be thrown out or combined according to some rule, to fit most of the available information into the frame. This may result in a smearing of some details.
TO DO BEFORE GOING TO COMPUTERS: Angular sizes are called "apparent" sizes because they depend on the distance to the object. Students can demonstrate the concept in the classroom by measuring any object of known size (including each other!) from different distances. One's fist, held at arm's length, is about 10 degrees. The physical height of the fist is measured to use in calculating actual heights.
Once the apparent sizes of the object have been recorded for different distances, they should be able to calculate the object height using the fact that the apparent size scales as the inverse of the distance: when the object is twice as far away, it has half the angular size.
Example: Yvonne's fist is 10 degrees at her arm's length of 23 inches. She also measures her fist to have a height of 3 inches. She makes a table like the following while measuring the height of a computer screen (about 14.5 inches). How can she compute its actual size?
Use the distance between Yvonne's eye and her fist as a basic unit of distance. Find the ratio of the arms' length to the distance from the object. Multiply the apparent size by this factor, and multiply by 3 inches for each 10 degrees.
d in × A deg × 3 in
Repeat for the different distances to see how well the measurements agree. Notice that all dimensions cancel, except inches, the units of the size, S. Since the measurements agree with one another but not with the screen's actual size, there is some systematic error. Probably the estimate of 10 degrees per fist is a bit inaccurate. If the distance measurements are fairly accurate, Yvonne can find the angular size of her fist (about 8.5 degrees) as homework.
The same principle is true in astronomy of course, but it is often much harder to measure the distances to objects. Parallax methods use opposite sides of the Earth's orbit as two observing sites. Distances out to about 3800 Parsecs have been measured (so far).
As an additional closure to the activity, have the students do a little research into galaxies their size, properties etc. , in particular looking up measurements of the distance to Andromeda.
|Answers to worksheet questions:
5a. Students should see a large, field of color with small scale variations and very few features. There is one bright spot in the upper left quadrant.
5b. Students may identify the one bright spot as Andromeda. The more perceptive may have doubts because the whole field is colored, indicating that something bright fills the field.
5c. The image spans 0.1 degree in 300 pixels, so each pixel is 0.1 / 300 = 3.33 × 10 -4 = .000333 deg/pix.
6a. This image shows about 2/3 of Andromeda's spiral galactic disk, along with two globular clusters that appear nearby. The background is visible in this linear-scale image as reddish areas in LL and UR, and background stars are scattered all over the image, including many between the observer and the galaxy, which appear against the outer disk.
6b. Andromeda occupies most of the image, and not all of it is included.
6c. Since the galaxy is diagonal with respect to the image borders, knowing the diagonal in pixel-widths will be handy:
The narrow dimension is about half this or 212 pixels, but we must remember that each pixel of this image is only one tenth the size we calculated in section 5.
An estimate using marks on the edge of a piece of paper indicates the long dimension is about twice the diagonal. Let's use scientific notation this time:
6d. The relative magnification of the second image is less than that of the first: 0.1x or one tenth.
6e. Students should realize at this point that what they saw in the highly magnified first image was the very center of the Andromeda galaxy.
7a. Now even the long dimension of Andromeda is smaller than the image diagonal, taking up about 3/4 of the distance. Since Andromeda's center is not quite along the image diagonal, this will have introduced some error in the calculations in 6.
7b. Changing the magnification doesn't change the image size in pixels, but it does change the angular size, or how much of the sky Andromeda seems to take up. The image diagonal is still about 424 pixels, but the magnification is reduced again by 1/3. Suppose the long dimension is now about 75% of the image diagonal:
This number is between an 8th and a 9th of the angular size at 1x1 degree. Compare this to the law of intensity, in which the amount of light crossing a set angular area decreases by the square of the increase in the distance. We have increased the apparent distance by a factor 3, so the size is about 1/9.
What is the real size of Andromeda? It can be calculated given its angular size and the magnification of an image, and the distance to the galaxy.
7c. We changed the apparent distance to the galaxy, and also its angular, or apparent size.
7d. The new magnification relative to 0.1 x 0.1 is 1/30.
7e. There are two other bright spots in the picture that look too bit to be stars. They are in fact globular clusters near the galaxy. (?)