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Astrophysics: Finding flux of a star given magnitude

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data

    The faintest stars a naked eye can see under the ideal conditions are of m = 6
    magnitude. Diameter of a maximally dilated pupil is d = 9mm. Calculate the
    magnitude of the faintest star a person can see if observing through binoculars
    (d = 5cm), a large backyard telescope (d = 8 inches), and a professional
    telescope (d = 2m). Compare these magnitudes to those found in part (a) -
    what kind of instrument would you need to see a Sun-like star at given distances?



    2. Relevant equations
    The magnitude equation:
    [itex]m-M=5log\left( \frac{d}{10} \right)[/itex]
    Where m and M are the apparent and absolute magnitudes of a star respectively. And the number 10 is in parsecs.

    3. The attempt at a solution
    My thoughts upon how to do this problem is to find the flux of the star that has a magnitude of 6, then see how much flux goes into a diameter of 9mm (pupil). From there I can relate that number to the amount of flux that goes into a backyard a professional telescope. The only problem is, it seems like I am not give enough information to find the flux! I probably have the wrong approach to this problem. Can anyone help? :\

    NOTE:
    This was part A, which I answered using the magnitude equation.
    Given that apparent magnitude of the Sun is m = 26:7 (at 1A:U:), nd its
    apparent magnitude at distances of 1pc; 10pc; 100pc and 1000pc.
     
  2. jcsd
  3. Feb 6, 2012 #2

    gneill

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    Staff: Mentor

    Perhaps you should consider the variation in brightness due to increased light-collecting ability of the binoculars over the naked eye. What's the ratio? How many magnitudes does this ratio represent?
     
  4. Feb 6, 2012 #3
    How can I get the brightness though? Is that the same as the flux? I cannot get a flux ratio because I do not know how far the magnitude 6 star is. :\
     
  5. Feb 6, 2012 #4

    gneill

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    Staff: Mentor

    You should be able to determine the flux ratio from the details of the light-collecting apparatus. How much more light does the binocular provide compared to the naked eye?
     
  6. Feb 6, 2012 #5
    I would imagine that the ratio of their surface areas, would that give the correct answer?

    So for the eye: [itex]S_e=\pi r^2= \pi (0.009)^2=2.54\times 10^{-4}m^2[/itex]
    And for the binoculars: [itex]S_b=\pi r^2= \pi (0.05)^2=0.15708m^2[/itex]
    So..
    [itex]\frac{S_b}{S_e}= \frac{0.15708}{2.54\times 10^{-4}}=617.284[/itex]

    But how do I relate this to a magnitude 6 star?
     
    Last edited: Feb 6, 2012
  7. Feb 6, 2012 #6

    gneill

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    Staff: Mentor

    You'll want to check your calculation for the binocular area. Also note that you've used the diameter values for radii -- which actually won't matter when the ratio is taken as the "error" then cancels out. You could save yourself a bit of calculator work by writing the ratio symbolically to start with and cancelling mutual constants before plugging in numbers.

    You should have in our notes or text a description that relates observed brightness ratios to magnitude differences. By how many magnitudes does the increased light-collecting ability of the binocular change the magnitudes of observed objects?
     
  8. Feb 6, 2012 #7
    The only equation that I have about this subject is simply the magnitude formula. I did something here and I am not sure if it is correct.

    I re-did the surface area ratio and got a more reasonable 30.86. I then plugged this in for the flux ratio in the magnitude formula..

    [itex]m_1-m_2=2.5log \left(\frac{F_2}{F_1} \right)[/itex]
    [itex]m_1-6=2.5log \left(30.86 \right)[/itex]
    [itex]m_1=9.72[/itex]

    Since 9.72 > 6 it is a fainter star. I do not know how sound this answer is though..
     
  9. Feb 6, 2012 #8

    gneill

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    Staff: Mentor

    Well, it looks quite reasonable to me. The binoculars add nearly four magnitudes to the "depth" of star that one can see.
     
  10. Feb 6, 2012 #9
    Same. I am going to do this as I feel this is correct. Thanks for your help! It is very much appreciated :]
     
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