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Same equation, different units

  1. Mar 12, 2005 #1

    tony873004

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    Starting fromthe equation [tex]\alpha=a/d[/tex] with [tex]\alpha[/tex] in radians and a and d in meters, show that the equation is also valid if [tex]\alpha[/tex] is expressed in arcseconds, a is in AU and d is in parsecs.

    Would this be the proper way to show this?

    [tex]\alpha=a/d[/tex]
    [tex]radians=meters/meters[/tex]
    [tex]4.8481*10^{-6} radians / arcsecond = \frac{1.49598*10^{11}m/AU}{3.0857*10^{16}m/pc}[/tex]

    Divide the numbers and cancel the m's



    [tex]4.8481*10^{-6} radians / arcsecond = 4.8481*10^{-6}AU/pc[/tex]

    Cancel the numbers
    [tex]radians / arcsecond = AU/pc[/tex]
    But radians is still there in the left part of the formula! What did I do wrong?
     
    Last edited: Mar 12, 2005
  2. jcsd
  3. Mar 12, 2005 #2

    dextercioby

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    U don't need any #-s.Just the definition of a parallaxis arcsecond:

    [tex] 1\mbox{parsec}=:\frac{1\mbox{AU}}{1\mbox{arcsecond}} [/tex]

    Daniel.
     
  4. Mar 12, 2005 #3

    tony873004

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    Thanks, Dex. The problem says we have to start with alpha in radians, a and d in meters and justify it that way. That's why I did it the way I did. I just don't know why the radians wont drop off, like the intuitive answer says they should.
     
  5. Mar 12, 2005 #4

    dextercioby

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    They do.Both radians & arcsecond are plane angle units...There's a connection between them

    [tex] 2\pi \ \mbox{radians}<--------------------->(180\cdot 3600) \ \mbox{arcseconds} [/tex]...

    Daniel.
     
  6. Mar 12, 2005 #5
    Just write down the identities:

    [tex] 4.8481 \cdot 10^{-6} \mbox{rad} = 1 \mbox{arcsecond}, \;
    1.49598 \cdot 10^{11} \mbox{m} = 1 \mbox{AU}, \;
    3.0857 \cdot 10^{16} \mbox{m} = 1 \mbox{parsec}, \;

    \Longrightarrow \frac{\mbox{arcsecond}}{4.8481 \cdot 10^{-6}} = 1 \mbox{rad} = \frac{1 \mbox{m}}{1 \mbox{m}} =
    \frac{\left(\frac{\mbox{AU}}{1.49598 \cdot 10^{11}}\right)}{\left(\frac{\mbox{parsec}}{3.0857 \cdot 10^{16}}\right)}[/tex]

    [tex]
    \Longrightarrow 2.0627 \cdot 10^{5} \mbox{arcsecond} = 2.0627 \cdot 10^{5} \frac{\mbox{AU}}{\mbox{parsec}}[/tex]

    [tex]
    \Longrightarrow 1\mbox{arcsecond} = \frac{1\mbox{AU}}{1\mbox{parsec}}
    [/tex]
     
    Last edited: Mar 12, 2005
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