The Global-Normal Disk Oscillations and the Persistent Low Frequency QPO in X-ray Binaries

We suggest that persistent low-frequency quasi-periodic oscillations (QPOs) detected in X-ray, ultraviolet, optical energy ranges the black hole (BH) sources XTE J1118+480, GRO J1655-40 LMC X-1 at ~ 0.1 Hz, and QPOs in HZ Her/Her X-1 at ~ 0.05 Hz and in Neutron Star (NS) binaries 4U 1323-62, 4U 1746-31 and EXO 0748-76 at ~ 1 Hz are caused by the global disk oscillations in the direction normal to the disk (normal mode). We argue that these disk oscillations are a result of the gravitational interaction between the central compact object and the disk. A small displacement of the disk from the equatorial plane results in a linear gravitational restoring force opposite to this displacement. Our analysis shows that the frequency of this mode is a function of the mass of the central object and it also depends on the inner and outer radii of the disk which in turn are related to the rotation period of the binary system. We derive an analytical formula for the frequency of the normal disk mode and show that these frequencies can be related to the persistent lower QPO frequencies observed in the NS and BH sources. We offer a new independent approach to the black hole mass determination by interpreting this low QPO frequency as the global disk oscillation frequency. The implementation of this method combined with the independent method which uses the X-ray energy spectra (Shrader & Titarchuk 1999) results in stringent constraints for the black hole masses.Comment: 14 pages, 1 figure, to appear in the Astrophysical Journal Letter

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.