  Elastic and proton-dissociative rho0 photoproduction (gamma p-->rho0 p,gamma
p -->rho0 N,with rho0-->pi+pi-) has been studied in ep interactions at HERA for
gamma-p centre-of-mass energies in the range 50<W<100 GeV and for |t|<0.5 GeV2;
the results on the p-dissociative reaction are presented for masses of the
dissociated proton system in the range MN^2<0.1W^2.For the elastic process,the
pi+pi- invariant mass spectrum has been investigated as a function of t. As in
fixed target experiments, the rho0 resonance shape is asymmetric;this asymmetry
decreases with increasing |t|.The cross section has been studied as a function
of W; a fit to the resonant part with the form W^a gives a=0.16\pm0.06(stat.)
+0.11-0.15(syst.). The resonant part of the gamma p-->pi+pi- p cross section is
11.2\pm 0.1(stat.)+1.1-1.2 (syst.) mub at <W>=71.7 GeV. The t dependence of the
cross section can be described by a function of the type Ae^(-b|t|+ct^2) with
b=10.9\pm0.3(stat.)+1.0-0.5(syst.)GeV-2 and c=2.7\pm0.9(stat.)+1.9-1.7(syst.)
GeV-4. The t dependence has also been studied as a function of W and a value of
the slope of the pomeron trajectory 0.23\pm0.15(stat.)+0.10-0.07(syst.)GeV-2
has been deduced. The rho spin density matrix elements r^04_00,r^04_1-1 and
Re[r^04_10] have been measured and found to be consistent with expectations
based on SCHC. For p-dissociative pi+pi- photoproduction in the rho0 mass
range, the distributions of the two-pion invariant mass, W and the polar and
azimuthal angles of the pions in the helicity frame are the same within errors
as those for the elastic process. The t distribution has been fitted to an
exponential function with a slope parameter 5.8\pm0.3(stat.)\pm0.5(syst.)GeV-2.
The ratio of the elastic to p-dissociative rho0 photoproduction cross section
is 2.0\pm0.2(stat.)\pm0.7(syst.).
