  We calculate the distance-dependent performance of a few representative
terrestrial neutrino detectors in detecting and measuring the properties of the
$\nu_e$ breakout burst light curve in a Galactic core-collapse supernova. The
breakout burst is a signature phenomenon of core collapse and offers a probe
into the stellar core through collapse and bounce. We examine cases of no
neutrino oscillations and oscillations due to normal and inverted neutrino-mass
hierarchies. For the normal hierarchy, other neutrino flavors emitted by the
supernova overwhelm the $\nu_e$ signal, making a detection of the breakout
burst difficult. For the inverted hierarchy (IH), some detectors at some
distances should be able to see the $\nu_e$ breakout burst peak and measure its
properties. For the IH, the maximum luminosity of the breakout burst can be
measured at 10 kpc to accuracies of $\sim$30% for Hyper-Kamiokande (Hyper-K)
and $\sim$60% for the Deep Underground Neutrino Experiment (DUNE).
Super-Kamiokande (Super-K) and Jiangmen Underground Neutrino Observatory (JUNO)
lack the mass needed to make an accurate measurement. For the IH, the time of
the maximum luminosity of the breakout burst can be measured in Hyper-K to an
accuracy of $\sim$3 ms at 7 kpc, in DUNE $\sim$2 ms at 4 kpc, and JUNO and
Super-K can measure the time of maximum luminosity to an accuracy of $\sim$2 ms
at 1 kpc. Detector backgrounds in IceCube render a measurement of the $\nu_e$
breakout burst unlikely. For the inverted hierarchy, a measurement of the
maximum luminosity of the breakout burst could be used to differentiate between
nuclear equations of state.
