Laboratory Exercise #3

The Heights of Lunar Craters

Please read the "Laboratory Rules of Operation" before proceeding.

Index

• Introduction & Goals
• Examine the images using XVista
• Collect auxiliary lunar data
• Measure the crater shadows
• Compute the heights of the crater walls
• Your Writeup
• Raw Data Files & Useful Links

This lab is based on an unfiltered CCD image of craters near the center of the first-quarter moon that were acquired with a SpectraSource Instruments Teleris CCD camera on the old 12-inch Tinsley telescope that used to reside in the dome on the roof of Smith Lab. The images were taken on 1998 September 28 EDT.

The Teleris camera uses a Kodak KAF-0400 768x512 CCD with 9-micron pixels. The Tinsley reflector was a 12-inch (0.305-meter) aperture telescope with an f/16 cassegrain focus. The CCD was roughly aligned with its long axis oriented along the East-West direction in the sky, and the short axis North-South. Both zero (bias) and flat-field calibration images were also acquired. Because the integration times are so short, no dark images were required.

Because we are viewing these craters just after dawn from their point of view, the crater walls cast long shadows. Using a simple geometric technique pioneered by Galileo Galilei in 1610, we can estimate the physical heights of the crater walls. Along the way we'll learn some basic image processing techniques.

Goals

The goals of this lab are as follows:

1. Learn to read in and display the image using the XVista package, and to make measurements of the apparent sizes of objects.

2. Learn how to identify a field using a finder chart (in this case, a photographic lunar atlas to identify the features in our CCD image), and to compile additional data (selenographic coordinates, and sun-earth-moon position information) from available catalogs (both books and online).

3. Using simple geometry, estimate the heights of crater walls from the lengths of their shadows, with uncertainty estimates, and compare these to typical measurements as available.

This lab will make use of the Linux workstations in MP4042 and the XVista image processing package. You will also need to use an Internet browser to search some online catalogs for information (either your own PC or one of the Windows PCs in MP4042), and you will need to use the reading room to consult copies of the various Lunar Atlases and The Astronomical Almanac for 1998.

[Index]

Login to one of the Linux workstations in MP4042. For this guide, I will denote Unix commands by a "%", which is the generic Unix command prompt. Commands in this guide given without a % are XVista commands (i.e., issued in response to the XVista "GO" prompt).

For this part, do the following:

1. Copy the raw data for this lab into your working directory by typing

       % cp /home/regulus/ast350/LabData/MoonLab/moon98.fits .
       

   If you type this command again, it will replace your working copies with new copies (and wipe out the old ones). It copies one (1) files into your working directory: moon98.fits, a pre-processed FITS-format image of the moon described above.

2. Startup the XVista program in an xterm window by typing

        % xvista
        

   once you get the XVista command prompt (GO:), you are ready to begin. For details on basic XVista commands, see the Vista Tutorial and Cookbook in the white binder in the lab workstation area.

3. Read in moon98.fits and display it. You may need to adjust the brightness and contrast using the L= and Z= keywords of the TV command. Adjusting the display contrast this way is far better than messing with the color bar at the bottom of the display (and less misleading). You can estimate the data values in the image by using the cursor and moving it over the image and noting the display of (x,y) position and "Intensity" in the box at the lower left-hand corner of the XVista TV Display window. This will help you chose sensible values of the zero-point ans span (Z and L, respectively).

4. Practice zooming in and out on features of interest (especially the craters), and measuring the coordinates of the centers and parts of the images (e.g., the crater walls and shadows). These kinds of measurements will be done in the steps below. This is your chance to explore the raw data and familiarize yourself with it before diving in to the more detailed analysis steps.

Extract Basic Image Information

Now that you have a reduced moon image to work with, you need to derive some basic information about this image. This includes

1. image size in pixels
2. image pixel scale (arcsec/pixel)
3. image field of view
4. minimum and maximum intensity values represented in the image.

Image Size

Display the image in XVista. What is the image size in pixels (e.g., size horizontally x size vertically)?

Pixel Scale

The Kodak CCD has a pixel size of 9-microns (square). The camera was located at the f/16 focus of the 0.305-meter Tinsley telescope. What is the pixel scale in units of arcseconds/pixels?

Image Field of View (FOV)

Combine your image size and pixel scale to compute the field of view of the image. Remember that the long axis is oriented East-West, and the short axis is oriented North-South. Express your FOV dimensions is both arcseconds and arcminutes.

Image Pixel Values

Use the abx command in XVista to analyze the pixel data values, as follows. If the image is in buffer 1, type

   abx 1 all
   

and read the output on the screen. Answer the following:

1. What is the minimum pixel value in the image? Which pixel is it located (give row,column coordinates)?

2. What is the maximum pixel value, and where is it located?

3. What is the average pixel value in the image?

Now, use the XVista histogram command to make a histogram showing the distribution of pixel data values in the image. Suppose that you found that the minimum and maximum pixels in your image were min=100 and max=1523, respectively (I made these up, your values will be different). If your moon image is in buffer 1, you would type

    histogram 1 xmin=100 xmax=1523 nolog
    

which will plot the histogram of pixel data values. Add the hard command to the line above to make a hardcopy and add this to your notes. Answer the following questions:

1. What is the mode of the pixel values (peak of the histogram)? Is it easy to define this quantity unambiguously in this image?

2. Label the mean you computed above on your plot of the pixel data value histogram. What do you think the "mean" represents in this particular image?

Make a hardcopy of the image using the IMPOST command. For example, to make a PostScript version of an image in buffer 1, displaying intensity values between 0 and 100 and pixel scale 0.54-arcsec/pixel (I made these up! - your values will be different!), type

    impost 1 z=0 l=100 scale=0.54 file=mycrater.ps positive
    

This will create a PostScript file with your image in the file mycrater.ps in your working directory. You should replace "mycrater" with your lastname, just so we know whose file it is. The positive keyword ensures that the picture comes out as a "positive" print.

To print the PostScript file from inside XVista use the command:

 
       $ lpr mycrater.ps
    

(or whatever you called the print file). The default printer is the laser printer in MP4042. As long as you are using the ast350 lab account, this printer will be the default printer (no guarantees for other accounts).

We are now ready for Part 2.

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Once you have located the field and identified the named craters (not all are in the Atlas), search the USGS Flagstaff Office website for the tables of Lunar Features (see under "Data and Software", then "Databases", then "Planetary Nomenclature", you follow the chain the rest of the way to a table of Lunar crater names. This gives a great table of the selenographic (lunar) latitudes and longitudes for all craters, as well as the diameter of the craters in kilometers, and other information that might be relevant or interesting.

"Seleno" is the genitive form applied to the moon, thus selenology is the lunar equivalent of terrestrial geology, etc. Hence, selenographic coordinates refer to a coordinate grid defined on the lunar surface.

Sun/Moon/Earth Configuration

Using the 1998 edition of The Astronomical Almanac look up the Ephemeris of Physical Observations of the moon to get the configuration of the Sun, Moon, and Earth (as seen from the Moon) at the time of our observation. You will need to interpolate the values in the table to our observation time (0113 UTC) on 1998 Sept 29 (UTC). For our observations, you need to get the following values from the table:

1. True distance of the moon (units are Earth radii). This is the distance between the centers of the Earth and Moon.

2. The Earth's selenographic longitude and latitude (point on the moon where the earth is on the zenith).

3. The Sun's selenographic co-longitude and latitude (read the notes to the table at the start of the section to see how co-longitude is related to longitude).

These data give us the critical viewing geometry for using the shadow lengths to estimate the heights of the crater walls.

Also look up values for the mean radius of the Earth and the Moon (in kilometers).

Annotated Image and Table

Annotate your hardcopy of the enhanced image with the names of the major craters, mountains and maria.

Compile a table of the craters you have identified, and give the data extracted from the sources described above. Also make an accompanying table of the date of observation and the sun/moon/earth configuration information that you derived from the sources above.

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Using XVista and your copy of the moon image, measure the lengths of the crater shadows for the two largest craters on your image. Your measurement should be in units of pixels. Do this as follows:

1. Display the enhanced image using the TV command.

2. Zoom in on the crater of interest.

3. Place the cursor at the center of the crater and measure the location of the center in (x,y) coordinates on the image.

4. Measure the length of the crater shadow by locating the (x,y) image coordinates of the leading and trailing edge of the shadow with the cursor. Denote the location of the leading edge as (x_l,y_l) and the trailing edge as (x_t,y_t). The length of the shadow, s will just be the geometric distance between the two:

   

   Make an estimate the uncertainties in your measurements, giving what assumptions you have made (think about what fraction of a pixel you can make measurements of the positions of features)..

5. Also measure the apparent diameters of the craters in a similar way.

6. Using the tabulated true distance of the moon (from The Astronomical Almanac as found above), and remembering that this is the distance between the centers of the Earth and Moon (and that we and the lunar features we are observing are on the respective surfaces), estimate the number of kilometers per arcsecond in our lunar images (i.e., how many km does 1 arcsecond correspond to).

7. Make a table of your measurements, giving the following:

   1. The pixel coordinates on your images of the crater centers

   2. The lengths of the crater edge shadows in pixels, arcseconds, and km.

   3. The apparent diameters of the craters in pixels, arcseconds, and km.

   Remember to include your estimates of the uncertainties of these measurements.

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The geometry of a crater shadow is shown in the figure below:

Fig. 1: Moon crater shadow geometry

The height of the crater is h, the apparent length of the shadow is s, and the altitude of the sun as seen from the end of the shadow is the angle a. They are related simply by:

Using these formulae, your measurements, and the selenographic coordinates of the Sun and craters interpolated from the tables, estimate the heights of the crater walls, and their uncertainties.

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Whew! That was involved, huh?

The research you had to do to hunt down everything was relatively simple (i.e., because I told you where to look), and is similar to what you encounter in any scientific project. Making the actual measurements is only one part of the problem. Making sense of the measurements, like using them to estimate something interesting (like the heights of lunar crater walls), is quite another.

For your writeup, try to organize all these various steps into a coherent narrative. However, also keep it brief: be concise and to the point. I won't be impressed by lots of extraneous information or off-topic speculation. I will, however, wonder why you haven't stuck to the topic.

A typical outline would be:

Introduction:

Briefly state the problem and what you intend to measure.

Observations:

Briefly describe the observations and the experimental data set that resulted from it (show the raw data, for example).

Data Reduction & Analysis:

Briefly describe how the measurements made from the data and how any measurement uncertainties were estimated. This is where you should give a table summarizing your measurements.

Results:

Summarize the additional data that were collected in a table and give the sources. Describe how you will go from measurements to derived quantities (in this case, describe the shadow projection geometry and how you go from observables to estimated crater heights, crater diameters, etc.). Present the results, and give "the answers", with uncertainties.

Discussion:

Summarize your final results, and interpret them. In this case, you might, for example, compare the heights you estimate for the one or two easily measured craters on your image to the heights of terrestrial impact features (e.g., Barringer Crater in Arizona). Are the different? Why?

Conclusions:

Wrap it up. Give a one or most two paragraph summary of the project, what you learned, what could be done better, etc.

This is not all that different than how one would organize observations, data analysis and interpretations into a scientific paper.

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An online catalog of lunar features, with selenographic coordinates and other data may be found at the website of the USGS Flagstaff Station in Flagstaff Arizona:

http://wwwflag.wr.usgs.gov/USGSFlag/

http://galilei.cmf.nrl.navy.mil/clementine/clib/features

Persons (other than Ast350 students who already have access to the data directly) are welcome to use the data and follow along with the lab. XVista is a publically available image processing package for Unix workstations (no Windows/NT/Mac ports planned), but any good image processing package that permits simple image arithmetic and smoothing operations will suffice. The trick is how to deal with the FITS image format.

The raw data consist of a single, unprocessed CCD image of the Moon acquired on 1998 Sept 29 01:13 (UTC) with the Teleris CCD camera on the 12-inch Tinsley Reflector on the roof of Smith Lab at OSU. A flat field and zero (bias) image were also acquired. This is the reduced, processed image:

moon98.fits (1.5M FITS Binary File)

rawmoon.fits (839k FITS Binary File)

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Copyright © Richard W. Pogge, All Rights Reserved.