Radial Mode Frequency Research Articles

Amplitude variability or close frequencies in pulsating stars -- the Scuti star FG Vir

The nature of the observed amplitude variability of several modes in the δ Scuti star FG Vir is examined. This is made possible by the extensive photometry obtained during 2002–04, as well as the long photometric time-base starting in 1992. In this star, three frequencies show strong amplitude and phase variations. In the power spectrum, these frequencies also show up as frequency doublets. However, since true amplitude variability of a single frequency can also lead to (false) frequency doublets in the power spectrum, a specific test examining in detail the observed amplitude and the phase variations of an assumed single frequency is applied. For the frequencies at 12.15 and 23.40 cycle d−1, it is shown that the amplitude variability of a single mode can be ruled out. In particular, an important property of beating between two modes is fulfilled: the amplitude and the phase vary synchronously with a phase shift close to 90°. The origin of the amplitude variability of a third mode, namely, near 19.86 cycle d−1, is not clear due to the long beat period of 20+ yr, for which the amplitude–phase test suffers from gaps in the coverage. However, even for this frequency, the amplitude variations can be expressed well by a mathematical two-mode model. If we examine these three close frequency pairs together with other (usually more widely separated) close frequencies in FG Vir, 18 pairs of frequencies with separations closer than 0.10 cycle d−1 have been detected. It is shown that the majority of the pairs occur near the theoretically expected frequencies of radial modes. Mode identifications are available for only a few modes: the only detected radial mode at 12.15 cycle d−1 is part of a close pair. It is shown that accidental agreements between the frequencies of excited modes can be ruled out because of the large number of detected close frequency doublets.

Squeezing carbon nanotubes

AbstractIn this paper we briefly review the high pressure Raman studies on single and double walled carbon nanotubes focussing on the perssure dependence of the low frequency radial breathing vibrational modes. Previous work on single walled carbon nanotube bundles have shown that the normalized pressure‐induced frequency shift, Φ = (1/ωR) (dωR/dP) ∼ D2, where ωR is the radial mode frequency and D is the nanotube diameter. We show here that the R‐modes due to the inner tubes in double walled carbon nanotubes, have a very small, diameter independent value for Φ, as predicted for the case of isolated tubes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Free Vibrations of an Orthotropic Thin Cylindrical Shell on a Pasternak Foundation

The basic objective of the present study is to investigate the influences of foundation parameters, orthotropy of materials represented by elastic moduli ratio, axial wave parameter, and circumferential wave number on three eigenfrequencies. Dynamic equilibrium equations and frequency equation are developed, and the eigenfrequencies are calculated. It is found that the foundation modulus primarily affects the radial mode eigenfrequency and has no effect on torsional and longitudinal modes, whereas the shear modulus does have influence on radial as well as tangential modes of vibration, though the effect on radial mode frequency is more dominating. The elastic moduli ratio chiefly affects the radial mode in a manner similar to that of a foundation modulus. When the shell rests on a Pasternak foundation, the lowest eigenfrequency vs circumferential wave-number characteristics do not have the well-known dip, and the minimum frequency occurs when the circumferential wave number is zero for all values of axial wave parameter.

Effect of van der Waals interactions on the Raman modes in single walled carbon nanotubes.

We have measured the Raman spectrum of individual single walled carbon nanotubes in solution and compare it to that obtained from the same starting material where the tubes are present in ordered bundles or ropes. Interestingly, the radial mode frequencies for the tubes in solution are found to be approximately 10 cm (-1) higher than those observed for tubes in a rope, in apparent contradiction to lattice dynamics predictions. We suggest that there is no such contradiction, and propose that the upshift is due rather to a decreased energy spacing of the Van Hove singularities in isolated tubes over the spacings in a rope, thereby allowing the same laser excitation to excite different diameter tubes in these two samples.

Suppressing g Modes in Shell Hydrogen-Burning δ Scuti Stars

For shell hydrogen-burning δ Scuti stars, many more unstable modes are predicted than are observed. For example, for 4 CVn and δ Scuti itself, only 18 and 6 modes are observed, respectively, whereas several hundred ℓ = 0, 1 and 2 rotationally-split modes are predicted. The predicted modes have a mixed p- and g-mode character, with many g-type nodes present in the H-exhausted core and shell H-burning region, where the Brunt-Väisälä frequency is large.Here we explore whether the predicted frequency spectrum can be made to agree better with the observed spectrum if the g-mode character of the pulsation modes is partially suppressed. Additional motivation for this approach is provided by the observation by Breger et al. (1999) that 4 modes in 4 CVn identified as ℓ = 1 have nearly equal frequency spacings of ∼14 μHz, reminiscent of consecutive radial orders from a pure p-mode spectrum. We consider 2.1- and 2.3-M⊙ evolutionary models that match the observed Teff, L, and log g, plus an identified radial mode frequency of 4 CVn and δ Scuti. For these models, we experimented with setting the Brunt-Väisälä frequency to zero from the model center to the edge of or beyond H-depleted core, so that the g-type portion of the waves becomes evanescent in this region.

Unstable non-radial modes in radial pulsators: theory and an example

We study the possibility of the excitation of non-radial oscillations in classical pulsating stars. The stability of an RR Lyrae model is examined through non-adiabatic non-radial calculations. We also explore stability in the presence of non-linear coupling between radial and non-radial modes of nearly identical frequency. In our model, a large number of unstable low-degree (ℓ = 1,2) modes have frequencies in the vicinity of unstable radial mode frequencies. The growth rates of such modes, however, are considerably smaller than those of the radial modes. We also recover an earlier result that at higher degrees (ℓ = 6–12) there are modes trapped in the envelope with growth rates similar to those of radial modes. Subsequently, monomode radial pulsation of this model is considered. The destabilizing effect of the 1:1 resonance between the radial mode and nearby non-radial modes of low degrees is studied, with the assumption that the excited radial mode saturates the linear instability of all other modes. The instability depends on the radial mode amplitude, the frequency difference, the damping rate of the non-radial mode, and the strength of the non-linear coupling between the modes considered. At the pulsation amplitudes typical for RR Lyrae stars, the instability of the monomode radial pulsation and the concomitant resonant excitation of some non-radial oscillation modes is found to be very likely.

The Free Vibration of a Cylindrical Shell on an Elastic Foundation

The frequency equation for a thin circular cylindrical shell resting on an elastic foundation is developed by using the first order shell theory of Sanders and eigenfrequencies are calculated. These eigenfrequencies are plotted against the axial wave parameter. Effects of the axial wave parameter, circumferential wave number, non-dimensional thickness and foundation parameters on eigenfrequencies are investigated. It is found that the foundation modulus chiefly affects the radial mode eigenfrequency and has no effect on torsional and longitudinal modes. On the otherhand, shear modulus does have influence on radial as well as tangential modes of vibrations. Though the effect on radial mode frequency is more pronounced.

The structure and oscillations of low-mass stars in the presence of a magnetic field

The equilibrium structure and oscillations of a partially degenerate standard model in the presence of a poloidal magnetic field have been studied. The magnetic field in the interior has been matched with an outside dipole field. The effect of magnetic field on the various structural parameters, e.g., mass, central condensation, moment of inertia, and oblateness has been computed for different values of the central degeneracy of the model. We have also studied the effect of magnetic field on radial oscillations of the configuration. A variational formulation is used to compute the changes in the frequency of radial mode of oscillation. It has been shown that the changes in frequency computed for various models using a two-parameter eigenfunction are in fair agreement with the values obtained by using the exact eigenfunction.

The effect of a magnetic field on the adiabatic oscillation of convective stellar models with radiation pressure

Using an unrestricted second order virial tensor analysis, the characteristic frequencies of small adiabatic oscillations of a convective stellar model in the presence of a weak poloidal magnetic field, with significant radiation pressure, have been calculated. In the absence of the magnetic field, the effect of increase in radiation pressure is to decrease the frequencies of radial modes, in spite of the increase in central condensation of the system. The characteristic frequencies of the transverse shear mode, the toroidal mode, and the pulsation modes show a general increase with the increase in the field strength. The marginal stability of the configuration in the presence of the magnetic field has also been discussed.

Harmonic properties of the annular membrane

Properties of a vibrating annular membrane are analyzed mathematically and numerically to investigate the relative harmonicity of the higher-order modes compared with the fundamental mode. The harmonicity depends on the zeros of a cross-product of Bessel functions. For a ratio of external-to-internal diameter up to about 1.2, the higher modes have frequencies which are very nearly exact integer multiples of the fundamental frequency. For circularly symmetric modes, the ratio of all higher radial mode frequencies to exact harmonic values is less than 1.0006 (a logarithmic cent), a ratio imperceptible to a listener. Angle-dependent modes also give good harmonicity, especially if some lower radial modes are suppressed. Thus the modes of an annular membrane whose external-to-internal diameter ratio is less than 1.2 are much more harmonic than those of the full circular membrane of a conventional drum. The theoretical harmonicity is at least as good as reported for stringed melodic instruments (piano, violin). The analysis suggests characteristics for construction of an annular drum-head which is supported on concentric rims of two cylinders; to sound the drum, a braced hoop might be used to excite radial modes, or strikers with grooved faces might be used to excite angular modes without exciting lower radial modes.