Using electromagnetic observations to aid gravitational-wave parameter estimation of compact binaries observed with LISA II: The effect of knowing the sky position

In this follow-up paper, we continue our study of the effect of using knowledge from electromagnetic observations in the gravitational wave (GW) data analysis of Galactic binaries that are predicted to be observed by the new \textit{Laser Interferometer Space Antenna} in the low-frequency range, $10^{-4} \:\mathrm{Hz}<f<1 \:\mathrm{Hz}$. In the first paper, we have shown that the strong correlation between amplitude and inclination can be used for mildly inclined binaries to improve the uncertainty in amplitude, and that this correlation depends on the inclination of the system. In this paper we investigate the overall effect of the other orientation parameters, namely the sky position and the polarisation angle. We find that after the inclination, the ecliptic latitude of the source has the strongest effect in determining the GW parameter uncertainties. We ascertain that the strong correlation we found previously, only depends on the inclination of the source and not on the other orientation parameters. We find that knowing the sky position of the source from electromagnetic data can reduce the GW parameter uncertainty up to a factor of $\sim 2$, depending on the inclination and the ecliptic latitude of the system. Knowing the sky position and inclination can reduce the uncertainty in amplitude by a factor larger than 40. We also find that unphysical errors in the inclinations, which we found when using the Fisher matrix, can affect the corresponding uncertainties in the amplitudes, which need to be corrected.