Teacher Page
Activity 8 Resolution

Standards (see Appendix A):

• Science Content Standards: A. Science as Inquiry, E. Science and Technology, G. History and Nature of Science.
• Mathematics Standards: Representation, Measurement, Connections.
• Technology Tools: Research; Problem Solving and Decision Making.

Unifying Concepts:  Systems; Evidence, Models and Explanation.

Objective: 

Students investigate images to recognize that resolution means sharpness of images and the ability to separate closely spaced objects. Students compare images to determine the relationship between telescope aperture and image resolution.  They create a graphical and mathematical model that demonstrates this relationship.  Finally, students learn that this relationship is limited for earth-based telescopes by seeing conditions caused by the atmosphere.  Students analyze star images using adaptive optics systems to see how image quality improves.

Overview:  

There are many factors that can affect the ability to distinguish objects in astronomical images.  Clouds, atmospheric turbulence, moisture, light pollution and the presence of the moon in the sky vary image quality.  Along with these external variables, there are factors related to the telescope optics that affect resolution.  In this activity students begin by learning how to quantify resolution using FWHM.  From there they employ the analysis techniques to uncover the relationship between aperture and resolution.  Finally students are introduced to the power of adaptive optics to control for atmospheric turbulence and improve resolution. 

In the introduction, students analyze a “slice line” of a star to determine the full width at half max (FWHM).  In section 8-a, students compare the slice graphs of three images of the ring nebula taken with three different aperture sizes.  In section 8-b, students apply this technique to images of Epsilon Lyrae (a double-double star system).  Again, the images were taken with the Yerkes 24 inch telescope with different amounts of the aperture covered.  A qualitative comparison reveals that two stars that are close together are more easily resolved with a larger aperture telescope, until seeing conditions become the limiting factor for resolution.  Students will describe the separation of the stars in two ways: by looking at the image and by using the slice tool to create graphs.  Students will collect data and quantitatively determine the relationship between resolution and the telescope’s aperture size. In section 8-c, students analyze a star image taken with adaptive optics systems turned off and turned on.

Background:

Vocabulary-

• Resolution – a measure of how sharp the image appears; the ability of a human eye, telescope or CCD camera to distinguish between closely spaced objects.
  
• Full Width Half Maximum (FWHM) - a measure astronomers use to describe resolution of star images. Under ideal conditions (no atmosphere and perfect telescope optics) all the light from a star should fall into a single pixel.  FWHM measures how much the light is spread out at half the maximum counts for the star.
  Using the slice tool on an image of a star produces a graph of the counts per pixel along the slice.    FWHM is a measure in pixels across the graph at one half the height of the peak.  The spread of a star’s light on an image can be quantified as FWHM.

Image Sets: 8 Resolution

• stars-fwhm  Image with many stars to slice and practice FWHM analysis.
• m57  Images of M57, the Ring Nebula in Lyra, taken with telescope apertures of 2 , 8, and 24 inches.  These are the same images explored in Activity 6 - Experimenting with Aperture. In this activity we are analyzing for resolution.  In Activity 6 we were analyzing for brightness.
• double-double  Images of the multiple star system, Epsilon Lyrae in Lyra.  Students analyze these images to discover resolving power indicated by the ability to separate two closely spaced stars.  Images taken with aperture masks. The first set includes images taken with apertures of 2, 4, 8, and 24 inches and exposure times compensating for aperture area, 13.5 sec., 3.38 sec., 0.84 sec., and 0.10 sec.  The second set, more-double-double has all images taken with the same exposure time, 0.10 seconds and images with taken with telescope apertures, 2, 4, 6, 8, and 24 inches.
• ao-on-off  Images of a star taken at Mt. Wilson Observatory with Adaptive Optics systems on and off. 

Introduction:  FWHM

        Instructions for showing and quantifying FWHM with HOU-IP software.

1.  Draw a short slice line through the center of a star on the image. The Slice Graph will pop up.
2. Choose Options, Axes Setup.
3. Adjust the Slice graph X axis. If the slice line is too long the numbers crowd together on the graph.  However one can limit the numbers shown on the X axis under the Options menu, Axes Setup.  Look at the Min and Max values for the X Axis.  Change the values to fit the part of the slice line which includes the star profile.  See example below of Slice graph for stars-fwhm.fts. Notice the X axis starts at 7 instead of 0, with the last value labeled as 19, though the slice extended beyond 19 in the original slice graph. 
4. Change the tick spacing on the X Axis to 1.
5. Adjust the Slice graph Y Axis.  Show the half-maximum and the peak line.    Look at the Min and Max values for the Y axis.  If Min is zero, just divide the Max value by two and enter the divided value in the tick spacing box.  If Min is not zero, then first subtract the value of Min from Max and then divide by two to determine the value for Tick spacing. Then look at the Max value for Y.  If the Max is an even number, add one to the value.  (This makes the peak line visible on the graph.)
6. Quantify FWHM. Count the distance in pixels across the graph at the half maximum point.  This distance is the number you record for FWHM. If you want to change the number to an angular size, multiply the number of pixels times the pixel scale in arcseconds.  See Activity 9 for more information on pixel scale.

Choose Options, Axes Setup, to control values for the slice plot to quantify the FWHM. 

Difference between min and max is the same as max in this case. 
But sometimes the min value is not zero.  See an example below for M57-2in_n90s.fts image.

Students should be able to calculate FWHM using any star.  All stars should produce the same FWHM for a given image at a given aperture.  So, student graphs should all have the same shape no matter which star they use for their calculation.  One way to show students that resolution is a characteristic of the entire image is to have them plot their graphs on overhead graph paper and overlay them.

Activity 8a  Calculating FHWM

Slice a Star.  In this activity students practice calculating FWHM using three images of the ring nebula, taken with a 2 inch, 6 inch, and 24 inch telescope aperture.  Students make a horizontal slice through each of the images, attempting to slice the pixels of greatest brightness.  For the graphic below the star chosen was the bright one to the left of the ring nebula.  Students are instructed to note the steepness of the curves as well as to calculate the FWHM of each. 

 

Quantify the slice graph.  Students should have learned how to quantify the slice graph in the above section.  The instructions for quantifying the slice graph are similar to the introduction activity, but worded slightly differently in this section.  These instructions include adjusting the style to include plot points and adjusting for a non-zero Y Min value.

Choose Options, Style.

1. Go to Options, Style on the slice graph.
2. Check plot the points, then click on Circle.

Options, Axes Setup.

1. Next go back to options, Axes Setup.
2. Click the box for grids for both the X and Y axes.
3. Change the tick marks to 1 for the X axis.
4. For the Y axes, change the value of the tick marks to half the difference between the minimum and the maximum value.
5. Finally, add 1 to the maximum value if it is an even number.  Click OK.
6. When the new display of the plot appears, use the mouse to grab the bottom right corner and stretch out the plot to see the numbers better.  You can shorten the display by changing the Y axis to show only the pixels of interest in the Y Axis setup.
7. Count the pixels at the midpoint.  The FWHM for the graph below would be the number of pixels crossed at half max, or about 5.75 pixels.  (For the Yerkes 24 inch telescope with the Apogee APy7p CCD, each pixel is 0.62 arcseconds.   So, the in arcseconds, the FWHM is 3.65  arcseconds.)

Activity 8b – The Double Double!

 

A telescope's resolution is limited by the aperture and the wavelength of light.

In 1835 George Bittle Airy determined that a mathematical relationship exists between telescope aperture and resolution.  The diffraction limit of a telescope is explained on a webpage of the Center for Adaptive Optics: http://cfao.ucolick.org/ao/   Following is an excerpt from the CfAO web page.

Under ideal circumstances, the resolution of an optical system is limited by the diffraction of light waves. This so-called "diffraction limit" is generally described by the following angle (in radians) calculated using the light's wavelength and optical system's pupil diameter: where the angle is given in radians. Thus, the fully-dilated human eye should be able to separate objects as close as 0.3 arcmin in visible light, and the Keck Telescope (10-m) should be able to resolve objects as close as 0.013 arcsec.

In the second part of this activity students calculate the FWHM of one of the pairs of stars in the Epsilon Lyrae system.  Students analyze the same stars for four different apertures and graph the results.  Images of Epsilon Lyrae were taken with the 24 inch telescope, covering the opening of the telescope to create 2 in., 4 in., 8 in., and 24 in. apertures.  Epsilon Lyrae is a double double in the constellation Lyra.  To most people it looks like one star.  With a telescope of adequate aperture, one can see two stars that each are doubles. A narrower star profile as indicated in the slice and quanitified as FWHM, indicates greater resolution. Resolution is proportional to telescope aperture. By analyzing this graph students can see that as aperture increases, FWHM decreases. However, after about 6 inches of aperture, atmospheric turbulence interferes with the improvement of resolution.

HINT: If students do not draw their slice through the brightest part of the double stars, they will have difficulty measuring the FWHM.  Students are instructed to increase the max brightness value adding two zeros to the Max value on the toolbar.  This will make the stars easier to distinguish.  In addition, using zoom and/or changing the color palette to a graded palette such as RAIN, will make the bright centers more obvious. The lower double is easier to analyze.

Activity 8c  – Adaptive Optics ON/OFF

The last part in this activity is a qualitative look at the importance of adaptive optics in improving resolution.   The Earth’s atmosphere has turbulence which moves around the star’s light as it approaches the telescope’s optics.  Exciting advances in telescope systems allow astronomers to change the shape of the mirror in response to measurements of the atmosphere’s turbulence.  As a result, much of the spreading out of the starlight is controlled resulting in sharper images.  Students compare the two images of a star, one taken with adaptive optics systems off and the other with the adaptive optics systems on.  Read more about adaptive optics projects at the website of the Center for Adaptive Optics.  http://cfao.ucolick.org/ao/

Homework:  To help students begin to think about resolution, you may want them to think about and construct a definition of resolution.

Time: 50 minutes to complete the activity, 30 minutes to discuss results.

Directions:

1. Ask the class to share their definitions of resolution.  Assist the group in clarifying that image quality is different than resolution.  The quality of an image is affected by seeing conditions, exposure time and proper focus to name a few factors. You can have a very good quality image that has relatively low resolution. Although issues of image quality can affect resolution, when we talk about resolution we are referring specifically to the ability to distinguish two objects from one another on the image.
2. Distribute the Student Page for this activity.  Emphasize that in this activity they will be learning a way to measure and compare the resolution of different images.  They will be then using this technique to uncover an important relationship between the telescope aperture and resolution.
3. Organize students to work in pairs for this activity.  Check their work as they finish the Introduction and 8a-FWHM to ensure that they are correctly calculating FWHM. If they are able to print, you may want to initial their printouts as they complete them. Otherwise, suggest student teams sketch the data graphs, adding their own interpretative explanations.

4. Interpretation and discussion of the results for Activity 8b-Double-Double, in particular, is key to student understanding.  Begin by comparing the slice plots they produced.  Consider making transparencies of their slice plots to share using an overhead projector, or exhibiting their sketches of the plots on poster size newsprint or butcher paper.
5. There is likely to be a great deal of variation in the student slice plots, depending on practice time, accuracy in slicing through the brightest pixels of each star, and consistency in the lengths of the slices.  Regardless of this variation, creating the slices and adjusting the options for display of their results will deepen student understanding of how astronomers analyze the resolution of telescope and imaging systems. 
6. For discussion purposes, you may choose to use one of these links showing screen shots of slices and the corresponding graphical plots. Link here to a page with Slice Plots.   More samples of slice plots, using the methods of using color palette, zoom box, slice and adjustment of plot options can be found on the page for Rainbow-Zoom-Slice-Plot.
7. From looking at a composite of all the slice graphs, students will again be able to conclude that as aperture increases, the peaks become more distinct.  These graphs show more clearly than the images alone the separation between the two stars. 
8. Students can graph the relationship between aperture and FWHM values, gathering data from their own slice plots or using the slice plots provided for the 2 inch, 4 inch, and 8 inch. Including the values for the 24 inch slice shows that once star images are resolved, increased aperture does not add much more to resolution.  This is because atmospheric 'seeing' dominates the results.  Removing atmospheric effects is an important goal of adaptive optics systems as applied to astronomy.
9. Assist students in comparing their analysis to the mathematical model established by George Airy.
10. Discussion of questions for Activity 8c-Adaptive Optics On / Off should lead to the realization of the profound effect that an adaptive optics system can have on image quality and resolution.  You may choose to have students visit the web site for the Center for Adaptive Optics for more details about how these system work.  http://cfao.ucolick.org/ao/  Students should realize after completing this lesson that a ground-based telescope equipped with adaptive optics and a larger aperture than Hubble Space Telescope should be able to produce images with greater resolution.  Economics limit the size of space based telescopes; it is expensive to launch space telescopes with mirrors of great size and weight.

Assessment/Evaluation:  Completed student pages or journal entries should reveal the significance of FWHM as a way to measure resolution, illustrate how FWHM is calculated, and include sketches or printouts in their descriptions. If students understand the concepts presented in this activity, they should be able to debate the advantages and limitations of large ground based telescopes, including the reasons for exploring adaptive optics systems.