Non phase-stabilised operation of PRIMA

In order to perform non phase-stabilised operation we must measure an observable which is not strongly affected by atmospheric phase changes. The simplest example is $C_{PS}'C_{SeS}'^{*}$:

$$C_{PS}'C_{SeS}'^{*} = C_{PS}A_{PS}\left ( t \right )\exp\left (i\theta\left ( t \right ... ...\left [\theta\left ( t \right )+\Delta \theta\left ( t \right )\right ]\right )$$ (13)

$$= C_{PS}C_{SeS}A_{PS}\left ( t \right )A_{SeS}\left ( t \right )\exp\left (-i\Delta \theta\left ( t \right )\right )$$ (14)

In the isoplanatic regime, $\left < \left \vert\Delta\theta\left ( t \right )\right \vert^{2}\right >_{t} <1$ and varies randomly about zero, so we can obtain an accurate measurement of $C_{PS}C_{SeS}^{*}$ by integrating $C_{PS}'C_{SeS}'^{*}$ over many fringe coherence times. $C_{PS}C_{SeS}^{*}$ is of great interest, as the phase of this number can be used to calculate the separation of the stars (the phase of $C_{PS}C_{SeS}^{*}$ is commonly called the astrometric phase, although it is the complex number $C_{PS}C_{SeS}^{*}$ which is the principle observable - a similar situation to the case of the bispectrum). The integration of $C_{PS}C_{SeS}^{*}$ can be improved by weighting each measurement of $C_{PS}'C_{SeS}'^{*}$ by an estimate of the S/N for that measurement.

In order to obtain a measurement of $C_{PS}'C_{SeS}'^{*}$ with good S/N in a reasonable period of time, both stars must provide a S/N ratio of $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}0.1$ for measurements of the correlated flux $C$.