wiki:astronomy:observational_astronomy:data_reduction_toa

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wiki:astronomy:observational_astronomy:data_reduction_toa [2025/01/07 14:16] – Roy Proutywiki:astronomy:observational_astronomy:data_reduction_toa [2025/02/05 20:33] (current) – Roy Prouty
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 ===== Data Reduction III: Atmospheric Extinction ===== ===== Data Reduction III: Atmospheric Extinction =====
  
-WRITE THIS +==== Recall the Pogson Equation==== 
- +From the definition of a [[wiki:astronomy:magnitude|magnitude]]
-==== Instrument Magnitude ==== +
- +
-Now invent an instrument-filter magnitude. +
- +
-Recall the definition of a [[wiki:astronomy:magnitude|magnitude]]+
  
 $$m_1 - m_2 = -2.5\log_{10}\frac{F_1}{F_2}$$ $$m_1 - m_2 = -2.5\log_{10}\frac{F_1}{F_2}$$
- 
-=== Observe that the collection of source #2 terms amounts to some constant === 
- 
-By properties of logarithms: $$m_1 - m_2 = -2.5\log_{10}F_1 - 2.5\log_{10}F_2$$ 
- 
-Grouping all source #2 terms: $$m_1 = -2.5\log_{10}F_1 + (m_2 - 2.5\log_{10}F_2)$$ 
- 
-Identify $m_2 - 2.5\log_{10}F_2 = C$ as an arbitrary constant. $$m = -2.5\log_{10}F + C$$ 
  
 ==== Atmospheric Extinction ==== ==== Atmospheric Extinction ====
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 $$dI = -\kappa(x)I(x)\sec{(z)}dx$$ $$dI = -\kappa(x)I(x)\sec{(z)}dx$$
  
-From here, we can note that $\frac{dI}{I} = d\ln{I}$ OR we can invent an integrating factor $\exp{(-\int\kappa(x)dx)}$. In either case, we will integrate over the path of the light ray: From the top of the atmosphere ($T$) to the bottom of the atmosphere, i.e., the ground ($B$).+From here, we can note that $\frac{dI}{I} = d\ln{I}$. We will integrate over the path of the light ray: From the top of the atmosphere ($T$) to the bottom of the atmosphere, i.e., the ground ($B$).
  
-=== With Differential Natural Logarithm === 
 Rearrange: Rearrange:
-$$\frac{dI}{I} = d\ln{I} = -\kappa(x)\sec{(z)}dz$$+$$d\ln{I} = -\kappa(x)\sec{(z)}dz$$
 Integrate: Integrate:
-$$\int_T^B\frac{dI}{I} = \int_T^Bd\ln{I} = \int_T^B-\kappa(x)\sec{(z)}dx$$+$$\int_T^Bd\ln{I} = \int_T^B-\kappa(x)\sec{(z)}dx$$
 We are left with: We are left with:
 $$\ln{I(x=T)} - \ln{I(x=B)} = -\sec{(z)}\int_T^B\kappa(x)dx$$ $$\ln{I(x=T)} - \ln{I(x=B)} = -\sec{(z)}\int_T^B\kappa(x)dx$$
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 $$\frac{I(x=T)}{I(x=B)} = e^{-\sec{(z)}K}$$ $$\frac{I(x=T)}{I(x=B)} = e^{-\sec{(z)}K}$$
  
 +==== Top-Of-Atmosphere Magnitude ====
 +From here, we must trust that $I \propto F$ in some way. See [[wiki:astronomy:radiative_quantities|Radiative Quantities]] if that trust is hard to come by.
  
 +Given this proportionality, we can say that $I_1$ is the top-of-atmosphere intensity and that $I_2$ is the intensity that we register on our detector (reduced to the top-of-telescope) intensity.
  
-----+$$m_T - m_B = -2.5\log_{10}\frac{I_T}{I_B}$$
  
-Sources+Given the results from above, this means that: 
 +$$m_T - m_B = -2.5\log_{10}e^{-\sec{(z)}K}$$
  
-  - [[https://slittlefair.staff.shef.ac.uk/teaching/phy217/lectures/principles/l04/]]+We can clean this up a little algebraically: 
 + 
 +$$m_T - m_B = -2.5(-\sec{(z)}K\cdot\log_{10}e)$$ 
 + 
 +$$m_T - m_B = 2.5\sec{(z)}K\cdot\log_{10}e$$ 
 + 
 +$$m_T = 2.5\sec{(z)}K\cdot\log_{10}e + m_B$$ 
 + 
 +So here is the work-up for the top-of-atmosphere magnitude based on $m_B$, $\sec{(z)}$, & $K$.\\ 
 +  - $m_B$ is the top-of-telescope instrument magnitude. 
 +  - $\sec{(z)}$ is the secant of the angle between zenith and the observed source 
 +  - $K$ is the atmosphere-integrated extinction 
 + 
 +The first two are easy (ish) ! That last one, the atmosphere-integrated extinction is less easy. It must be derived either from measurements of the layer-by-layer extinction coefficient and numerically integrated OR we must find some way to measure the atmosphere-integrated extinction itself. 
 + 
 +==== Measuring Atmospheric Extinction ==== 
 + 
 +Take a look at the final result from above again: 
 +$$m_T = 2.5\sec{(z)}K\cdot\log_{10}e + m_B$$ 
 +Choose to rearrange this in the form of a standard $y=mx+b$ linear form with $m_B$ as the dependent variable: 
 +$$m_B = -K\Biggl(2.5\sec{(z)}\cdot\log_{10}e\Biggr) + m_T$$ 
 +With this, we can see that on a graph of $m_B$ vs $2.5\sec{(z)}\cdot\log_{10}e$, the magnitude of the slope will be $K$ and the y-intercept will be $m_T$.\\ \\ 
 + 
 +Therefore, in order to measure $K$, we must observe a source for which $m_T$ is constant throughout the session and we must observe this source at a variety of zenith-angles, $z$. The top-of-telescope instrument magnitudes resulting from enough observations at differing zenith angles should trace a line. A fit to that line will result in a y-intercept of the top-of-atmosphere magnitude for the observed source and a slope that can serve as a reasonable estimate of the atmosphere-integrated extinction.\\ \\ 
 + 
 +It's important to to note that the source we aim to understand as a part of the overall observing stands to be different from the source we observe to measure the extinction. Further, the source we observe to measure extinction should be non-variable in nature and appear in the same region of the sky as the source in question. This source we observe the measure extinction should be called the "extinction star". 
 + 
 + 
 + 
 +----
  
-{{tag>[not-done]}}+{{tag>[done]}}