### Determination of Stellar Magnitudes from Digital Images

OBJECTIVES

• Learn and understand stellar magnitude concept.

• Practice calculating stellar magnitudes with spreadsheet.

• Calculate intrinsic magnitudes of stars based on distance from the star.

Note: We will do Procedure I on one day, then return later to Procedure II to calculate magnitudes.

INTRODUCTION. Astronomers describe the apparent brightness of stars in magnitudes (see Kaufmann, p. 227f). Hipparchus developed the magnitude scale in ancient Greece. Arbitrarily, he said that a 6th magnitude star was 100 times fainter than a 1st magnitude star. A magnitude difference means a certain ratio in intensities. Let r be the ratio of intensities for stars differing by one magnitude. r x r = r2 is the ratio of intensities for stars differing by two magnitudes, etc. r5 = 100. Thus,

The magnitude scale is logarithmic, i. e. the magnitude of a star is related to the exponent of the intensity. Another strange characteristic of the magnitude scale is that it is backwards. A magnitude 6 star is 100 times fainter than a magnitude 1 star. Thus magnitude is a logarithmic measure of "faintness".

Suppose we have a standard star of magnitude mo which yields an intensity reading of Io from a photometer. Another star reads I on the same photometer. A simple formula for the magnitude m is given by

m = m0 - 2.5*log(I/I0)

Although this formula is rather cryptic involving logarithms, it presents no problem for a calculator or computer spreadsheet.

PROCEDURE I. Measure brightness with digital imaging. The class has previously photographed a number of stars of varying brightness using the CCD digital imaging system.  Click here for a list of the stars for 2002-2003.  You will load digital images of these stars on machines in the Spidel Computer Laboratory using Matlab.

1. Start Matlab using the start button.

2. When given the Prompt inside Matlab, type colorphot.  Colorphot is a stand-alone module inside Matlab written by Don Collins.  Use pushbuttons and the mouse to load images, subtract dark frames, and measure brightness.

3. Load a star image.  This is the [Load Webcam] button.  In the dialog box, you need to navigate to the directory on G:  G:\classes\physics\astronomy\StarColor\031016class\ and G:\classes\physics\astronomy\StarColor\031113class\   There may be another directory in the path.  The last numbers are the date on which the stars were recorded.

4. Another pushbutton menu appears to change the contrast and visibility.  Try different settings until you are satisfied with the image, then press [OK]. Be sure to leave the contrast adjust before proceding.

5. Note the exposure time that is coded in the filename for that star.

6. Measure the star red intensity.  [Measure color].  This will give both blue brightness and red brightness and the ratio of blue/red.  For the magnitude measurements, you need only record the red brightness.

7. Record the red brightness for each star on the list.  Also record the total exposure time from the time code in the file name.  Example: "_25secx10" means 10 frames at 1/25 sec each for a total exposure time of .04 sec x 10 = 0.4 sec.

The computer adds all the pixels in a small circle centered on the cursor and subtracts the average sky brightness in an annular region surrounding the circular aperture. In case the mouse position is not accurate, the computer tries to optimize the brightness count by moving the aperture around in a small area of the image until it finds the largest brightness signal. You should repeat the brightness for each star a couple of times to verify that you obtain the same numbers each time. If your cursor is too far from the star’s image, you will get meaningless results or a non-number: NaN.

Make a table in your notebook with multiple columns:

 Star name and filename Exposure time (shutter time x no. of frames) Red Brightness (ADU) Red Intensity (ADU/sec) (Brightness/Expos time) Magnitude

The brightness (in ADU) is the consistent number obtained from the Bright function.

Be sure to make a complete table in your notebook, not on this sheet. The columns should contain "Star name", "File name", "Red brightness (ADU)", "Exposure time (sec)". Calculate the star brightness in ADU/sec by dividing image brightness by exposure time. Notice the difference between intensity and brightness.  The intensity accounts for the various exposure times.  Make a list of all the measured stars from brightest to dimmest and compare with your "eyeball" ordering.  Include this list in your summary.

PROCEDURE II Use spreadsheet to calculate the magnitudes. After the table of the image brightness and exposure times has been completed, enter all the data into a table in a spreadsheet. The spreadsheet is located at: http://www.warren-wilson.edu/~physics/Contemp-Astronomy/starmag/StarMag.xls This spreadsheet has been all set-up with starnames, star distances, exposure times as the file memo, etc. The star names are in alphabetical order according to constellation. All computers on campus can run the spreadsheet.

Each student should complete the spreadsheet according to the directions below:

1. Calculate the exposure time in sec using the “memo”. For example, “1/25 x 20” is 20/25 sec = .8 sec. Enter the calculated exposure time in sec in the appropriate column. Do not include the units, just the numbers.

2. Enter the image brightness for each star. This is the data obtained from the Matlab “Colorphot” procedure. Some stars have more than two images, and some stars were photographed on more than two nights. If so, you may insert new rows for additional star images. In this case, you should enter the star name and other data. Save your spreadsheet file frequently!

3. Calculate the Image Intensity for each star. Here you will program the spreadsheet to make the calculation. Press “=” to get into calculation mode and point to the cells and enter the arithmetic operators to enter the formula. Watch the example in class. Enter the formula only in the top cell in the column. Use the “drag-dot” to replicate the formula for all elements in the column. Watch the demo.

4. Calculate the star magnitude for each star using the logarithm formula. m = m0 – 2.5*log(I/I0) We use the magnitude of Altair for m0 = 0.8. I0 is the intensity of Altair. Pay close attention to the class demonstration of how to point to the cells for Io using what is refered to as “Absolute addressing” and the <F4> function key. Replicate the formula for all stars, and check magnitudes against the published magnitudes. Your calculated data should agree with the published magnitudes within 0.2 mag. Some data may be incorrect such as a mis-labeled star file.

5. Calculate the intrinsic magnitude (the magnitude as would be measured from a distance of 32.6 LY (10 pc)). Do this in a new column and enter and replicate the formula for intrinsic magnitude:

Where d is the distance in LY (the distance column). This formula uses the “inverse-square” relation, but is applied to logarithms.

1. (Optional) Delete empty rows. Some of the stars in the spreadsheet template have no observed data due to time constraints. You may delete these rows – especially since the calculations give error messages such as divide by zero.

2. Save for later. Be sure to save your spreadsheet for the next experiment where we will add the color analysis.

3. Summarize the data. When you are sure that the spreadsheet is correct, print-it out (be sure to have your name on the top). Make a table in your notebook sorting the stars by apparent magnitude. Make another table in your notebook sorting the stars by intrinsic magnitude. (If you are brave, you may invoke sorting functions in the spreadsheet). Discuss the results, and any surprises you observe. Be sure to discuss the meaning of intrinsic magnitude and the reason why astronomers want to express star brightness in “intrinsic magnitudes”.