LISA Pathfinder platform stability and drag-free performance

The science operations of the LISA Pathfinder mission have demonstrated the feasibility of sub-femto-g free fall of macroscopic test masses necessary to build a gravitational wave observatory in space such as LISA. While the main focus of interest, i.e., the optical axis or the $x$-axis, has been extensively studied, it is also of great importance to evaluate the stability of the spacecraft with respect to all the other degrees of freedom (d.o.f.). The current paper is dedicated to such a study: the exhaustive and quantitative evaluation of the imperfections and dynamical effects that impact the stability with respect to its local geodesic. A model of the complete closed-loop system provides a comprehensive understanding of each component of the in-loop coordinates spectral density. As will be presented, this model gives very good agreement with LISA Pathfinder flight data. It allows one to identify the noise source at the origin and the physical phenomena underlying the couplings. From this, the stability performance of the spacecraft with respect to its geodesic is extracted as a function of frequency. Close to 1 mHz, the stability of the spacecraft on the ${X}_{\mathrm{SC}}$, ${Y}_{\mathrm{SC}}$ and ${Z}_{\mathrm{SC}}$ d.o.f. is shown to be of the order of $5.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{\ensuremath{-}2}\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ for $X$, $6.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{\ensuremath{-}2}\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ for $Y$, and $4.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{\ensuremath{-}2}\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ for $Z$. For the angular d.o.f., the values are of the order of $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\text{ }\text{ }\mathrm{rad}\text{ }{\mathrm{s}}^{\ensuremath{-}2}\text{ }\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ for ${\mathrm{\ensuremath{\Theta}}}_{\mathrm{SC}}$, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{rad}\text{ }{\mathrm{s}}^{\ensuremath{-}2}\text{ }\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ for ${H}_{\mathrm{SC}}$, and $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{rad}\text{ }{\mathrm{s}}^{\ensuremath{-}2}\text{ }\text{ }{\mathrm{Hz}}^{\ensuremath{-}1/2}$ for ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{SC}}$. Below 1 mHz, however, the stability performances are worsened significantly by the effect of the star tracker noise on the closed-loop system. It is worth noting that LISA is expected to be spared from such concerns, as differential wave-front sensing, an attitude sensor system of much higher precision, will be utilized for attitude control.