1.  What is the ratio of intensities for a pair of stars 5 magnitudes apart? 
    10 magnitudes? 15 magnitudes? 20 magnitudes? 
    Is there a simple rule for calculating these intensity ratios quickly?

         (m1 - m2) = 5  mag   -->   I2  =            100 I1

         (m1 - m2) = 10 mag   -->   I2  =         10,000 I1

         (m1 - m2) = 15 mag   -->   I2  =      1,000,000 I1

         (m1 - m2) = 20 mag   -->   I2  =    100,000,000 I1

         Every difference of 5 mag means another factor of 100
         in intensity ratio.


2. The average diameter of the dark-adapted pupil in a human eye 
   is about 6 millimeters; the average person can see a star of 
   magnitude 6 on a clear, dark night.  If the same person were 
   to look through typical 7x35 binoculars, how faint a star might 
   he be able to detect?

        eye diameter = 6 mm   -->  eye radius    R1  =  3 mm
      
                                   eye area      A1  =  pi * R1^2 =  28 sq. mm


     binocular diam  = 35 mm  ->   binoc radius  R2  = 17 mm

                                   binoc area    A2  =  pi * R2^2 = 908 sq. mm

         
        binoc area       908 sq. mm
       ------------  =  ------------  =  32 times larger
        eye area          28 sq. mm


     Therefore, binocular user can see stars 32 times fainter than eye user.
     The ratio of intensities corresponds to a mag difference

                                          Area(binoc) 
      m(binoc) - m(eye)  =  2.5 * log10 ( ------------ )
                                          Area(eye)


                         =  2.5 * log10 ( 32 )

                         =  2.5 * 1.5

                         =  3.8


     So, if the faintest magnitude visible to eye is m(eye) = 6, then

               m(binoc)  =  3.8 +  6

                         =  9.8




3. My eyesight is so poor that I can't distinguish Alcor from Mizar 
   without my eyeglasses. Given magnitudes of 4.00 for Alcor 
   and 2.06 for Mizar, what is the magnitude of the single 
   blurry object I see?

       First, we convert from magnitude to intensity.  Let I(M) be
       intensity of Mizar, and I(A) be intensity of Alcor.

           I(A)        0.4 * ( 2.06 - 4.00 )
         -------  =  10                        =   0.167
           I(M)
       

       That means that if I add together the intensities, I get

          I(M)  +  I(A)  =  I(M)  +  0.167*I(M)

                         =  1.167*I(M)

       So the combined intensity is 1.167 times the intensity of
       Mizar alone.  I can now figure out the magnitude of this
       combined blur.
                                              I(M)
        m(combined) - m(M)  =  2.5 * log( ----------- )
                                          I(combined)

                                              I(M)
        m(combined) - m(M)  =  2.5 * log( ----------- )
                                           1.167 I(M)

                                               1  
        m(combined) - m(M)  =  2.5 * log( ----------- )
                                             1.167      

                            =  2.5 * ( -0.671 ) 

                            =  -0.168

      And so

        m(combined)         =  -0.168  +  m(Mizar)

                            =   1.89



4. What is the magnitude of star HD 1964? 

       Go to SIMBAD   http://simbad.u-strasbg.fr/sim-fid.pl and find

               mag (HD 1964)  =  10.14  in B filter
                                  9.77  in V filter