Excitation of Trapped g-Mode Oscillations in Warped Disks around Black Holes

arXiv:1004.0555v2 [astro-ph.HE] 27 Apr 2010 Excitation of Trapped g-Mode Oscillations in Warped Disks around Black Holes Finny oktariani Department of Cosmoscience, Graduate School of Science, Hokkaido University, Kita-ku, Sapporo 060-0810, Japan finny@astro1.sci.hokudai.ac.jp Atsuo T. okazaki Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605, Japan and Shoji kato 2-2-2 Shikanodai-nishi, Ikoma-shi, Nara 630-0114, Japan (Received 2009 0; accepted 2010 0) Abstract In order to study the origin of high-frequency quasi-periodic oscillations observed in X-ray binaries, Kato (2004) suggested a resonant excitation mechanism of disk oscillations in deformed disks. In this paper, we investigate numerically, following his formulation, whether trapped g-mode oscillations in a warped disk, where the warp amplitude varies with radius, can be excited by this mechanism. For simplicity, we adopt Newtonian hydrodynamic equations with relativistic expressions for the char- acteristic frequencies of disks. We also assume that the accretion disk is isothermal. We find that the fundamental modes of trapped g-mode oscillations with eigenfre- quencies close to the maximum of epycyclic frequency are excited. The intermediate oscillations found are isolated in a narrow region around the resonance radius. After varying some parameters, we find that the growth rate increases as the warp ampli- tude or the black hole spin parameter increases, while it decreases as the sound speed increases. Key words: accretion, accretion disks – black holes – high-frequency quasi- periodic oscillations – relativity – stability 1. Introduction The Rossi X-ray Timing Explorer (RXTE) satellite has detected high-frequency quasi- periodic oscillations (QPOs) in X-ray binaries. They are kilohertz (kHz) and hectohertz (hHz) QPOs in neutron-star low-mass X-ray binaries, and high-frequency (HF) QPOs (≥100 Hz) in 1 black-hole X-ray binaries. HF QPOs occur at fixed frequencies and the appearance is correlated to the state of the sources, that is, they appear only in high-luminosity states where L > 0.1LE, with LE being the Eddington luminosity. They are suggested to be phenomena originating in a strong gravitational field, which occur in the innermost region of the relativistic disk. HF QPOs are regarded as being a powerful tool to explore the mass and spin of the central black hole, and also to explore the physical states of the innermost region of the accretion disk. Various models have been proposed to explain HF QPOs. For example, Abramowicz & Klu´zniak (2001) and Klu´zniak & Abramowicz (2001) pointed out the importance of resonant processes as being the cause of HF QPOs. In their model, HF QPOs are the result of resonant couplings between the vertical and horizontal epicyclic oscillations at a particular radius. There is, however, uncertainty concerning the excitation process of the oscillation system as a whole. Kato (2004; 2008a; 2008b) proposed a model where HF QPOs are regarded to be disk oscillations resonantly excited in deformed disks. The deformation considered is a warp or an eccentric disk deformation in the equatorial plane. An outline of the model is as follows. A non-linear coupling between a disk oscillation (hereafter, an original oscillation) and a deformed part of the disk (warp or eccentric deformation) causes some forced disk oscillations (hereafter, intermediate oscillations). The intermediate oscillations make a resonant coupling with the disk, and then feedback to the original oscillation. Since the nonlinear feedback process involves a resonance, the original oscillation is amplified or damped. There are two kinds of resonances, i.e., horizontal resonance (Lindblad resonance) and vertical resonance. When the deformation is a warp, the resonance that can excite p- and g-mode oscillations is the horizontal resonance (Kato 2004; 2008a; 2008b). Hence, in this paper we consider the horizontal resonance alone. In the Keplerian disks this resonant process works only when the disk is general relativistic. That is, a non-monotonic radial distribution of epicyclic frequency is necessary for the appearance of resonance and for trapping of oscillations. Recently, Ferreira & Ogilvie (2008) generalized Kato’s idea on the excitation of trapped g-mode oscillations and made detailed numerical calculations of the growth rates of the oscillations in the case of rotating black holes. Their results indicate that the coupling mechanism can provide an efficient excitation of trapped g- mode oscillations, provided that the global deformations reach the inner part of the disk with non-negligible amplitude. In this paper, we consider nearly the same problem examined by Ferreira & Ogilvie (2008), following the formulation by Kato (2008a; 2008b). The purpose of this paper is to make comparison of results obtained from two different approaches, Ferreira & Ogilvie’s fully numerical approach and Kato’s analytical one. We especially consider t

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