  This paper is concerned with the critical threshold phenomenon for one
dimensional damped, pressureless Euler-Poisson equations with electric force
induced by a constant background, originally studied in [S. Engelberg and H.
Liu and E. Tadmor, Indiana Univ. Math. J., 50:109--157, 2001]. A simple
transformation is used to linearize the characteristic system of equations,
which allows us to study the geometrical structure of critical threshold curves
for three damping cases: overdamped, underdamped and borderline damped through
phase plane analysis. We also derive the explicit form of these critical
curves. These sharp results state that if the initial data is within the
threshold region, the solution will remain smooth for all time, otherwise it
will have a finite time breakdown. Finally, we apply these general results to
identify critical thresholds for a non-local system subjected to initial data
on the whole line.
