  We investigate the limitations of thermonuclear X-ray bursts as a distance
indicator for the weakly-magnetized accreting neutron star 4U 1728-34. We
measured the unabsorbed peak flux of 81 bursts in public data from the Rossi
X-Ray Timing Explorer (RXTE). The distribution of peak fluxes was bimodal: 66
bursts exhibited photospheric radius expansion and were distributed about a
mean bolometric flux of 9.2e-8 erg/cm^2/s, while the remaining (non-radius
expansion) bursts reached 4.5e-8 erg/cm^2/s, on average. The peak fluxes of the
radius-expansion bursts were not constant, exhibiting a standard deviation of
9.4% and a total variation of 46%. These bursts showed significant correlations
between their peak flux and the X-ray colors of the persistent emission
immediately prior to the burst. We also found evidence for quasi-periodic
variation of the peak fluxes of radius-expansion bursts, with a time scale of
approximately 40 d. The persistent flux observed with RXTE/ASM over 5.8 yr
exhibited quasi-periodic variability on a similar time scale. We suggest that
these variations may have a common origin in reflection from a warped accretion
disk. Once the systematic variation of the peak burst fluxes is subtracted, the
residual scatter is only approximately 3%, roughly consistent with the
measurement uncertainties. The narrowness of this distribution strongly
suggests that i) the radiation from the neutron star atmosphere during
radius-expansion episodes is nearly spherically symmetric, and ii) the
radius-expansion bursts reach a common peak flux which may be interpreted as a
standard candle intensity.Adopting the minimum peak flux for the
radius-expansion bursts as the Eddington flux limit, we derive a distance for
the source of 4.4-4.8 kpc.
