Derivation

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

Doppler shift:

(1)
\begin{align} \frac{\delta v}{c} = \frac{\delta \lambda}{\lambda} \end{align}

Assumption Doppler shift is small compared to line width.

Observable intensity change at a given pixel:

(2)
\begin{eqnarray} F \left ( i \right) -F_0 \left ( i \right) &=& \frac{\partial F_0 \left ( i \right)}{\partial \lambda} \delta \lambda \left ( i \right)\\ &=& \frac{\partial F_0 \left ( i \right)}{\partial \lambda}\frac{\delta v\left ( i \right)}{c} \lambda \left ( i \right) \end{eqnarray}

Doppler shift restated:

(3)
\begin{align} \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 ]} \end{align}

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

(4)
\begin{align} W\left ( i \right) = \frac{1}{\left ( \frac{\delta v_{RMS}\left ( i \right)}{c}\right)^2} \end{align}
page revision: 2, last edited: 17 Sep 2007 20:06