**Publication date:** Apr 2013

**Abstract:**

Context. Using known frequencies of the twin-peak high-frequency

quasiperiodic oscillations (HF QPOs) and known mass of the central black

hole, the black-hole dimensionless spin a can be determined by assuming

a concrete version of the resonance model. However, a wide range of

observationally limited values of the black hole mass implies low

precision of the spin estimates. Aims: We discuss the possibility

of higher precision for the black hole spin a measurements in the

framework of a multi-resonance model inspired by observations of more

than two HF QPOs in the black hole systems, which are expected to occur

at two (or more) different radii of the accretion disc. This framework

is also applied in a modified form to the neutron star systems.

Methods: We determine the spin and mass dependence of the twin-peak

frequencies with a general rational ratio n:m, assuming a non-linear

resonance of oscillations with the epicyclic and Keplerian frequencies

or their combinations. In the multi-resonant model, the twin-peak

resonances are combined properly to give the observed frequency set. For

the black hole systems we focus on the special case of duplex

frequencies, when the top, bottom, or mixed frequency is common at two

different radii where the resonances occur giving triple frequency sets.

Results: The sets of triple frequency ratios and the related spin

a are given. The resonances are considered up to n = 5 since excitation

of higher order resonances is improbable. The strong resonance model for

"magic" values of the black hole spin means that two (or more) versions

of resonance could occur at the same radius, allowing cooperative

effects between the resonances. For neutron star systems we introduce a

resonant switch model that assumes switching of oscillatory modes at

resonant points. Conclusions: In the case of doubled twin-peak HF

QPOs excited at two different radii with common top, bottom, or mixed

frequency, the black hole spin a is given by the triple frequency ratio

set. The spin is determined precisely, but not uniquely, because the

same frequency set could correspond to more than one concrete spin a.

The black hole mass is given by the magnitude of the observed

frequencies. The resonant switch model puts relevant limits on the mass

and spin of neutron stars, and we expect a strong increase in the

fitting procedure precision when different twin oscillatory modes are

applied to data in the vicinity of different resonant points. We expect

the multi-resonance model to be applicable to data from the planned LOFT

or similar X-ray satellite observatory.

Appendices are available in electronic form at http://www.aanda.org

**Authors:**

Stuchlík, Z.; Kotrlová, A.; Török, G.;