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Computing luminosity from surface brightness

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Homework Statement


I'm trying to find the central luminosity per square parsec of a galaxy with central surface brightness [tex]I(0) = 15 \; mag \; arcsec^{-1}[/tex]. I need the answer to be in multiples of the solar bolometric luminosity per square parsec.

Homework Equations


[tex]m_1 - m_2 = 100^{\frac{1}{5}}\log{\left(\frac{F_1}{F_2}\right)} = (2)\left(100^{\frac{1}{5}}\log{\left(\frac{d_1}{d_2}\right)}\right)[/tex]
(the formula for absolute magnitude follows easily be letting [tex]d_2 = 10 \; pc[/tex])


The Attempt at a Solution


I know that surface brightness is independent of distance (for nearby objects, at least). Since 1 arcsec spans a distance of 10 A.U at [tex]d = 10 \; pc[/tex], finding the galaxy's magnitude per square parsec should be easy (computed to be 3.5236E-14). But if I try to find the Sun's magnitude per square parsec - [tex]\frac{4.83}{4 \pi R_{sun}^2}[/tex] where [tex]R_{sun}[/tex] is expressed in parsecs - I get very large number. This leads me to believe that my reasoning itself is faulty.

What is the correct way to approach this problem?
 
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