wiki:astronomy:observational_astronomy:observational_astronomy:extinction

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wiki:astronomy:observational_astronomy:observational_astronomy:extinction [2023/11/14 13:02] – Roy Proutywiki:astronomy:observational_astronomy:observational_astronomy:extinction [2023/11/14 13:11] (current) – Roy Prouty
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 So blazing forward with this equivalence, we can re-write the Pogson Equation in terms of this atmospheric extinction relationship we've just derived. So blazing forward with this equivalence, we can re-write the Pogson Equation in terms of this atmospheric extinction relationship we've just derived.
  
-$$m_b-m_t = -2.5 \log_{10}\frac{I_\lambda^b}{I_\lambda^t}$$+$$m_b-m_t = -2.5 \log_{10}\frac{I_\lambda^b}{I_\lambda^t} = -2.5 \log_{10}\Biggl[\exp{\Biggl(- \sec{z} \int_b^t \alpha_\lambda(x) dx\Biggr)}\Biggr]$$ 
 + 
 +Let's carefully take the log-base-10 of the exponential... 
 +$$m_b-m_t = -2.5 \log_{10}(e)\biggl(- \sec{z} \int_b^t \alpha_\lambda(x) dx\biggr)$$ 
 + 
 +Finally, recalling that we want to calibrate to the top-of-the-atmosphere, we find 
 + 
 +$$m_t = m_b + 2.5 \log_{10}(e)\biggl(- \sec{z} \int_b^t \alpha_\lambda(x) dx\biggr)$$ 
 + 
 +So we find that the top-of-the-atmosphere magnitude is our bottom-of-the-atmosphere = top-of-telescope magnitude with some additional magnitude that was "extincted" away from incidence as the radiance traversed the atmosphere.\\ \\ 
 + 
 +Note that the $F$ or $I$ used in the above equations are //not// counts. They must be some unit of power. So radiance, flux, energy per second (power), or even counts-per-second. 
 + 
 +---- 
 +Sources 
 +  - Lifted with minor modification from: https://slittlefair.staff.shef.ac.uk/teaching/phy241/lectures/l07/