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Getting Optimal Weight from a Spectrum

To derive the equation for the information one can get from a spectrum.

Doppler shift:

\displaystyle  \frac{\delta v}{c} = \frac{\delta \lambda}{\lambda}

Assuming that the Doppler shift is small compared to absorption line width, the observable intensity change at a given pixel:

\displaystyle  F \left ( i \right) -F_0 \left ( i \right) =  \frac{\partial F_0  \left ( i \right)}{\partial \lambda} \delta \lambda \left ( i \right)

\displaystyle  F \left ( i \right) -F_0 \left ( i \right) = \frac{\partial F_0  \left ( i \right)}{\partial \lambda}\frac{\delta v\left ( i \right)}{c} \lambda \left ( i \right)

Doppler shift restated:

\displaystyle  \frac{\delta v\left ( i \right)}{c} = \frac{F \left ( i \right) -F_0 \left ( i \right) }{\lambda\left ( i \right) \left [ \partial F_0 \left ( i \right) / \partial \lambda \left ( i \right)\right ]}

Add over available spectrum. Each pixel contributes according to the optimal weight:

\displaystyle  W\left ( i \right) = \frac{1}{\left (  \frac{\delta v_{RMS}\left ( i \right)}{c}\right)^2}

citation: http://adsabs.harvard.edu/abs/2007PhRvL..99w9001M

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