2 Exact solution for (piecewise) linear source function

Source function

$\begin{displaymath} S(\tau) = S_0 + \frac{\Delta S}{\Delta \tau } \left( \t... ...e \enspace \mbox{in the interval} \enspace [ \tau_0 , \tau_1 ] \end{displaymath}$

(2)

and the abbreviations

$\begin{displaymath} \Delta \tau := \tau_1 - \tau_0 \enspace , \enspace \enspac... ... S_1 - S_0 \enspace , \enspace \enspace \Delta I:= I_1 - I_0 \end{displaymath}$

(3)

give for the change of the intensity

$\displaystyle \Delta I\left( I_0, S_0, S_1, \Delta \tau \right)$

$\textstyle =$

$\displaystyle \left( I_0 -S_0 \right) \left[ e^{- \Delta \tau } -1 \right] \... ...} \left[ e^{- \Delta \tau } - \left( 1 - \Delta \tau \right) \right] \enspace .$

(4)

Special care has to be taken for optically (very) thin and thick regimes.