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Optical Powers
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Optical Powers
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Standards (see Appendix A):
Unifying Concept: Evidence, Models and Explanations, Size and Scale
Objective:
Students understand that pixel scale is an angular measure of sky subtended by the telescope/CCD system and is dependent upon the actual size of the pixels and the focal length of the telescope. Students learn to calculate this scale in arcseconds for any combination of telescope and CCD.
Purpose:
The field of view for a telescope/CCD and the pixel scale are really different sides of the same coin. When considering field of view, you want to know how large a slice of the sky is able to fit onto the chip of your CCD camera. Pixel scale is simply a smaller version of the same problem – measuring the slice of sky that fits onto one pixel of your CCD chip. Talking about field of view is most useful when you are asking the question: Will the object I am interested in fit into the space I have available on the CCD chip? Pixel scale, on the other hand, is useful once you have taken an image to make calculations regarding the size of different features in the image. The calculation for each is the same. The application is slightly different. We begin this activity with pixel scale.
Drawing upon what they have learned in previous activities regarding focal length and angular size, students are lead through the steps that allow them to calculate the amount of sky that will fit onto a particular CCD chip. Students are first asked to recall the effect that increasing focal length has on field of view and image scale or magnification. They are then introduced to pixel scale as a relationship of the size of the pixel with the magnification of the image falling on the CCD chip.
Teacher Background:
Two things are going to affect the amount of sky your telescope/CCD system and, consequently, each pixel of the CCD chip are going to be able to capture on an image: the focal length of your telescope and the size of the pixels. In Activity 4 – F Box Explorations, we discovered the effect of changing focal length. Recall that as focal length increases, the image becomes more magnified and the field of view becomes smaller. The CCD chip then records the image produced by the telescope. The larger the CCD chip is the more of the image that will be recorded. So, it is the combination of focal length and size of the recording surface that determines the actual field of view for any system.
When you look up the size of a particular target, it is often expressed in terms of its angular size. Because celestial objects appear very small, you will most often see these angle measurements expressed in smaller units of arcminutes (') and arcseconds (").
360 degrees in a circle
60 minutes in a degree
60 seconds in a minute
360° in a circle
60' in a 1°
60" in 1’
We use these same units to express the amount of sky that fits on each pixel – pixel scale. However, you can’t just look this number up in the CCD owner’s manual because pixel scale depends upon the magnification of the image sent to the CCD along with the size of the pixel itself. We can look up the actual size of the pixel. We can also find out the focal length of the telescope we are using. Using these values, it is a matter of a little math to find the pixel scale.
Pixel scale = 3438 arcminutes x pixel size / focal length
Pixel scale = 206,265 arcseconds x pixel size / focal length
Inquiring minds want to know! What are the 3438 and the 206265 doing in the equations?
The Short Answer – Dividing pixel size by focal length results in an answer in radians. There are 57.3° in one radian, so there are 57.3 x 60 arcminutes per radian. Multiplying by 3438 converts radians to arcminutes. There are 57.3 x 3600 arcseconds in each radian. Multiplying by 206,265 converts radians to arcseconds. For the purpose of this activity where we are not attempting to determine the actual size of the object, just the angular size, it is more convenient to have answers in arcseconds or arcminutes.
The Long Answer – A radian is an angle measurement with no units. It is equal to the angle subtended by an arc on a circle that is equal in length to the radius of that circle. Because of this relationship between s and r, the following equation is true:
s = r x q when q is measured in radians.
Construct a Radian. You need a circular plate and pipe cleaners. Using a pipe cleaner, measure the radius of a circle. Cut a bunch of pipe cleaners to the length of the radius. Take that length and lay it along the circumference of the circle. Tape it down. Lay another pipe cleaner along the length and tape it down, continuing all the way around the circle. Now take two of the pipe cleaner radii and tape them from the center of the circle to each end of one of the radii lengths along the edge of the circle. The measure of the enclosed angle is called a radian. Measure it with a protractor.
Think about it! How many radii lengths of pipe cleaner could you fit along the circumference of the circle? ___ Did it turn out even or was there a bit of the circle left over? Does this make sense considering that the angular measure of a radian is equal to about 57.3 degrees? Explain.
A radian is the measure of the angle theta (θ) that subtends an arc of length s equal to the radius r of a circle. When r = s the angle θ is equal to 1 radian.
There are 2
radians in a circle; one radian is equal to 57.29578°. Or, 3438 arcminutes, or 206,265 arcseconds.
Because the field of view for any pixel or the entire chip is much smaller than one radian, we are allowed to use what is called the small angle approximation. In the small angle approximation equation, the angle θ becomes so small that D approaches equality with s. Referring to the figure below, we also make D the diameter of an astronomical object (moon, sun, star, etc.) and we change r to d (distance to the object).
