  This work characterises the sky localization and early warning performance of
networks of third generation gravitational wave detectors, consisting of
different combinations of detectors with either the Einstein Telescope or
Cosmic Explorer configuration in sites in North America, Europe and Australia.
Using a Fisher matrix method which includes the effect of earth rotation, we
estimate the sky localization uncertainty for
$1.4\text{M}\odot$-$1.4\text{M}\odot$ binary neutron star mergers at distances
$40\text{Mpc}$, $200\text{Mpc}$, $400\text{Mpc}$, $800\text{Mpc}$,
$1600\text{Mpc}$, and an assumed astrophysical population up to redshift of 2
to characterize its performance for binary neutron star observations. We find
that, for binary neutron star mergers at $200\text{Mpc}$ and a network
consisting of the Einstein Telescope, Cosmic Explorer and an extra Einstein
Telescope-like detector in Australia(2ET1CE), the upper limit of the size of
the 90% credible region for the best localized 90% signals is
$0.25\text{deg}^2$. For the simulated astrophysical distribution, this upper
limit is $91.79\text{deg}^2$. If the Einstein Telescope-like detector in
Australia is replaced with a Cosmic Explorer-like detector(1ET2CE), for
$200\text{Mpc}$ case, the upper limit is $0.18\text{deg}^2$, while for
astrophysical distribution, it is $56.77\text{deg}^2$. We note that the 1ET2CE
network can detect 7.2% more of the simulated astrophysical population than the
2ET1CE network. In terms of early warning performance, we find that a network
of 2ET1CE and 1ET2CE networks can both provide early warnings of the order of 1
hour prior to merger with sky localization uncertainties of 30 square degrees
or less. Our study concludes that the 1ET2CE network is a good compromise
between binary neutron stars detection rate, sky localization and early warning
capabilities.
