# 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:

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