To date, 17 circumbinary planets have been discovered. In this paper, we focus our attention on the stability of the Kepler circumbinary planetary systems with only one planet, i.e. Kepler-16, Kepler-34, Kepler-35, Kepler-38, Kepler-64 and Kepler-413
. In addition to their intrinsic interest, the study of such systems is an opportunity to test our understanding of planetary system formation and evolution around binaries. The investigation is done by means of numerical simulations. We perform numerical integrations of the full equations of motion of each system with the aim of checking the stability of the planetary orbit. The investigation of the stability of the above systems consists of three numerical experiments. In the first one we perform a long term (1Gyr) numerical integration of the nominal solution of the six Kepler systems under investigation. In the second experiment, we look for the critical semimajor axis of the six planetary orbits, and finally, in the third experiment, we construct two dimensional stability maps on the eccentricity-pericentre distance plane. Additionally, using numerical integrations of the nominal solutions we checked if this solutions were close to the exact resonance.
We present a novel high resolution contactless technique for thermal conductivity determination and thermal field mapping based on creating a thermal distribution of phonons using a heating laser, while a second laser probes the local temperature thr
ough the spectral position of a Raman active mode. The spatial resolution can be as small as $300$ nm, whereas its temperature accuracy is $pm 2$ K. We validate this technique investigating the thermal properties of three free-standing single crystalline Si membranes with thickness of 250, 1000, and 2000 nm. We show that for 2-dimensional materials such as free-standing membranes or thin films, and for small temperature gradients, the thermal field decays as $T(r) propto ln(r)$ in the diffusive limit. The case of large temperature gradients within the membranes leads to an exponential decay of the thermal field, $T propto exp[-A cdot ln(r)]$. The results demonstrate the full potential of this new contactless method for quantitative determination of thermal properties. The range of materials to which this method is applicable reaches far beyond the here demonstrated case of Si, as the only requirement is the presence of a Raman active mode.
The specific heat of ultra-thin free-standing membranes is calculated using the elastic continuum model. We first obtain the dispersion relations of the discrete set of acoustic modes in the system. The specific heat is then calculated by summing ove
r the discrete out-of-plane wavevector component and integrating over the continuous in-plane wavevector of these waves. In the low-temperature regime (T < 4 K), the flexural polarization is seen to have the highest contribution to the total specific heat. This leads to a linear dependence with temperature, resulting in a larger specific heat for the membrane compared to that of the bulk counterpart
