  We report on variations in important X-ray emission lines in a series of
Chandra grating spectra of the supermassive colliding wind binary star Eta
Carinae, including key phases around the X-ray minimum/periastron passage in
2003.5. The X-rays arise from the collision of the slow, dense wind of Eta Car
with the fast, low-density wind of an otherwise hidden companion star. The
X-ray emission lines provide the only direct measure of the flow dynamics of
the companion's wind along the wind-wind collision zone. We concentrate here on
the silicon and sulfur lines, which are the strongest and best resolved lines
in the X-ray spectra. Most of the line profiles can be adequately fit with
symmetric Gaussians with little significant skewness. Both the silicon and
sulfur lines show significant velocity shifts and correlated increases in line
widths through the observations. The R = forbidden-to-intercombination ratio
from the Si XIII and S XV triplets is near or above the low-density limit in
all observations, suggesting that the line-forming region is >1.6 stellar radii
from the companion star. We show that simple geometrical models cannot
simultaneously fit both the observed centroid variations and changes in line
width as a function of phase. We show that the observed profiles can be fitted
with synthetic profiles with a reasonable model of the emissivity along the
wind-wind collision boundary. We use this analysis to help constrain the line
formation region as a function of orbital phase, and the orbital geometry.
