Floquet analysis of two-dimensional perturbed Keplerian flows in cataclysmic variables

Dipartimento di Astronomia, Università di Padova, vicolo dell'Osservatorio 3, 35122 Padova, Italy e-mail: fabrizio.tamburini@unipd.it

Received: 19 March 2010
Accepted: 2 May 2010

Abstract

Aims. We apply Floquet analysis to study the dynamical stability of two-dimensional axisymmetric Keplerian flows distorted by an m = 2 epicyclic periodic forcing term, representing a tidal perturbation in a thin accretion disk (AD) of a cataclysmic variable (CV). This tidal mode is responsible for the disk truncation mechanism.

Methods. Floquet analysis reveals, with a positive Floquet exponent, exponentially growing modes of the local instabilities that can cause the disruption of the AD. The equations of the inviscid fluid used to model the AD describe an oversimplified two-dimensional tidally-distorted accretion disk. Viscosity and the other properties expected in a CV AD are taken from the Shakura-Sunyaev α-disk solution and introduced in the model through the thermodynamical quantities present in the equations. In particular, we address this investigation in search of parametric resonances of the radial Fourier components of the velocity field in the multi-dimensional parameter space of the solutions describing a fluid stream in a CV AD: the mass of the primary M₁, the orbital period p of the system, the orbital fiducial radius r of the fluid stream and the normalized Fourier mode of the radial motion, k.

Results. We find that, in region of the r3:2 orbital resonance radius, where superhumps (SH) form, the m = 2 tidal epicyclic modes alone would not introduce any perturbation in this region of the AD, and therefore the Floquet exponent of the m = 2 tidal force is found to be either negative or zero. However, only by introducing the 3:1 tidal perturbation, will the dynamics of the fluid streams at the SH radius become unstable indicating the onset of SH. Finally, Floquet analysis of the Fourier decomposition of the derivative along the radial x-direction shows the important role of the low Fourier modes of the radial velocity perturbations in the disk truncation mechanism.