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Bright accretion-powered systems exhibit many varieties of time-variabilty, including flaring and quasiperiodic oscillations. These may reflect variations in the source itself intrinsic to rapid accretion; to assess this idea we examine the time-dependent response of radiation-dominated spherical accretion flows to radial and nonradial perturbations. Such flows may serve as useful first approximations to the actual accretion flows in low-mass X-ray binaries (LMXBs) and other systems. Feedback between the production of radiation and the flow of gas to the radiation producing regions promotes the existence of global modes, each of which is associated with a characteristic oscillation about steady flow. The feedback mechanism is very general, implying that radiation hydrodynamic oscillations may be a universal feature of rapidly accreting systems. These oscillations are accompanied by characteristic variations in the radiation output of the accretion flow. Because heavily damped modes are unlikely to be observable, we examine how the frequencies and the damping rates of global modes depend on parameters such as the mass accretion rate. Under conditions thought to be typical of LMXBs, the radial modes are weakly damped and some nonradial modes are even linearly unstable. Our results for the frequencies and structure of radial modes confirm similar findings by Fortner, Lamb, and Miller~(1989). When nonradial modes are active, the flow develops regions through which material preferentially accretes, and regions through which radiation preferentially escapes. Our linear stability analysis suggests that these modes dominate the accretion flow behavior when the system luminosity approaches the Eddington limit, the critical point beyond which steady spherical accretion becomes impossible. Nonradial mode frequencies behave very differently from those of the radial modes, as they appear to depend only weakly on the system luminosity. A connection may exist between nonradial radiation hydrodynamic modes and the NB/FB quasi-periodic intensity oscillations found in LMXBs, which also display striking frequency stability.
