• Learn and understand stellar magnitude concept.
• Practice calculating stellar magnitudes with spreadsheet.
• Calculate intrinsic magnitudes of stars based on distance from the star.

INTRODUCTION. Astronomers describe the apparent brightness of stars in magnitudes (see Comins p. 179f). 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, 

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

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 a digital SLR camera.  Cassiopeia, Fall, 2008.   You will load digital images of these stars on machines in the Spidel Computer Laboratory or Physics Laboratory using Matlab.  Matlab is a sophisticated and powerful analytical package that is programmable.  In future years, a photometry package that runs without the expensive Matlab program may be available. 

Click here to examine a small portion of a digital star image.

We will measure selected stars in the constellation Cassiopeia - photographed Oct. November 6, 2008 by Gracie McCarroll, Chandler Jones, and Bayle Owens.

• Examine the image :Cassiopeia, Fall, 2008.  Write down the star name that you think is the brightest star by examining the negative image.  2nd brightest star?  Faintest of the labeled stars?  We will then measure the brightnesses.
• Start Matlab.  This is a big program, please be patient!
  
• From Matlab, just type starbright.  This is an application developed by Donald Collins to measure the brightness of stellar images by "point and click" techniques with the user's mouse.  The camera used to make the images is a digital SLR camera using raw images.  Raw images display a linear light response.
  
• starbright asks for a file.  If you are on WWC campus, navigate to G:\physics\Astronomy\2008Cas_Owens_McCarroll_Chandler and select any of the three images that end in ".fit".  starbright loads all three B, G, and R images, and it doesn't matter which one the user selects.  These images were made by adding 20 digital images at 15 sec each.  Total exposure time of 10 x 15sec = 300 sec. 
  
• Adjust the contrast to your satisfaction.  Use the contrast control on the image window.  Adjust the contrast so that the faint stars are visible, but the image doesn't look too noisy.  You may want to try a Negative image.  You can re-adjust the contrast and pos/neg at any time throughout the session.
• NOTE:  DO NOT ENTER NUMBERS FOR THE DISTANCE, OR CALIB STAR .  DO NOT PUSH THE "ENTER CALIB STAR MAG" BUTTONS.  THEY ARE FOR ANOTHER PURPOSE AND WILL INTERFERE WITH TODAY'S MEASUREMENTS.
  
• Take the mouse and click on a star in the image.  Note: In order to view the image better and see the stars better, you should zoom up with 1 click of the zoom button.  Then you will have to pan the image to get each star in the field of view.  To calculate the brightness of a star, the computer reads the sky background in an annular region surrounding the star.  The average background is subtracted.  Then the net value of all the pixels in a small aperture that includes the star are added to give the brightness in ADU's.  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.  The brightness is displayed in another window as Analog Digital Units: ADU.  If the user misses a star by such a large amount that the search algorithm fails,  nothing is displayed.  The user should try clicking more accurately.
  
• Record the starbrightness in a table in your notebook along with the star name, distance, and Star brightness (ADU).  See this link for an image of Cassiopeia (2008) and the star names and distances.  ADU means "Analog Digital Units" and represents the sum of all the pixels added up in the aperture (as was done with the pseudostar lab).
• When finished with all the stars, press the "Done" button in the star image.  You should also exit Matlab and log off the computer.
  

• Save your work!

• Which star is the brightest apparent magnitude?  Which star is the second brightest?  Which star is the faintest? How do these compare with your estimates at the beginning?
• Which star is the brightest Intrinsic magnitude?  Which star the second brightest? Which star the faintest?
  
• Save your work!
• Calculate the Luminosity relative to the sun.  The luminosity of a star is the total power radiated from the star.  It is convenient to compare the luminosity in solar units.  If a star has a luminosity of 10 solar units, that star is 10 times brighter than the sun when the sun and the star are viewed from the same distance.  The intrinsic magnitude of the sun is 4.83.   For a star with an intrinsic magnitude of M, the luminosity in solar units is given by thr formula:

Where 4.83 is the intrinsic magnitude of the sun.  In this formula, (4.83 - M)/2.5 is the exponent, or the power of 10.  This is also taking the antilog of a number.  When typing this strange formula in the spread sheet, you have to write it in the following format:  10^((4.83-M)/2.5).  Use the pointing feature of the spreadsheet to point to the intrinsic magnitude cell.  You must be very careful with the parentheses.  Note that first you subtract numbers, then divide the difference by 2.5, then rais 10 to the result!  The brightest star should be larger than 30,000 times brighter than the sun.  The "^" operator means "raise to a power".

• Which star has the largest luminosity?  How many times brighter than the sun is the brightest luminosity?
• Which star has the smallest luminosity?  Which stars emit less total light then the Sun?
  
• Print the spreadsheet.  Make sure the spreadsheet is correct and your name is at the top.
• Optional: Sort the spreadsheet before printing in order of apparent magnitudes and again in order of luminosity.  You need to be careful here.  Be sure to select all the data, and only the data, before sorting.  Make sure all rows become sorted.  You should certainly save the spreadsheet before sorting so a mess-up can be restored by reloading the saved file.
  

Cassiopeia

Star Distance (LY) alpha 229 beta 54.5 gamma 613 chi 204 delta 99 epsilon 442 eta 19.4 kappa 4129 lambda 355 mu 24.6 theta 137 upsilon1 229 upsilon2 206 zeta 597