Constraints on the tidal dissipation factor of a main sequence star: The case of OGLE-TR-56b

The planet OGLE-TR-56b is the extrasolar giant planet closest to its host star. This planet and its star exchange extreme tidal forces. This leads to a reduction of the planetary orbit and a spin-up of the stellar rotation. The tidal migration rate depends crucially on the ratio of the tidal dissipation factor $Q_{*}$ and the stellar love number $k_{2{,}^{*}}$of the star, which is only poorly known and estimates range within $5\times 10^5<(Q_{*}/k_{2^{*}})<10^{10}$. For values greater than $Q_{*}/k_{2^{*}}>1.5\times 10^9$no observable influence by tidal forces on the planet's orbit within the lifetime for the star can be found. A lower limit for the possible values of the parameter $Q_{*}/k_{2^{*}}$for the G-type star OGLE-TR-56 was found by studying the evolution of possible tidal interaction into the future and in the past. This study demonstrates that on the basis of conservative model assumptions, a considerable but unrealistic spin-up of the star can be expected if $Q_{*}/k_{2^{*}}<2\times 10^7$, which is not in agreement with observed stellar rotation periods. From a statistical analysis based on a Monte-Carlo tidal evolution simulation, the $Q_{*}/k_{2^{*}}$ parameter can be constrained to the range $2\times 10^7<Q_{*}/k_{2^{*}}<1.5\times 10^9$ if the system shall evolve significantly and realistically by tidal forces.     

New intrinsic-colour calibration for uvby-β photometry

A new intrinsic-colour calibration $((b-y{)}_o-\beta )$ is presented for the uvby-β photometric system, making use of re-calibrated Hipparcos parallaxes and published reddening maps. This new calibration for $(b-y{)}_o-\beta$, our Eq. (1), has been based upon stars with $d_{\mathit{Hip}}<70\text{pc}$ in the photometric catalogues of Schuster and Nissen (1988), Schuster et al., 2004, Schuster et al., 2006, provides a small dispersion, $\pm 0.009$, and has a positive “standard” $+2.239\mathrm{\Delta }\beta$ coefficient, which is not too different from the coefficients of Crawford (1975a, +1.11) and of Olsen (1988, +1.34). For 61 stars with spectra from CASPEC, UVES/VLT, and FIES/NOT databases, without detectable Na I lines, the average reddening value $〈E(b-y)〉=-0.001\pm 0.002$ shows that any zero-point correction to our intrinsic-colour equation must be minuscule.     

An application of the tensor virial theorem to hole + vortex + bulge systems

The tensor virial theorem for subsystems is formulated for three-component systems and further effort is devoted to a special case where the inner subsystems and the central region of the outer one are homogeneous, the last surrounded by an isothermal homeoid. The virial equations are explicitly written under the additional restrictions: (i) similar and similarly placed inner subsystems, and (ii) spherical outer subsystem. An application is made to hole + vortex + bulge systems, in the limit of flattened inner subsystems, which implies three virial equations in three unknowns. Using the Faber-Jackson relation, $R_{\text{e}}\propto {\sigma }_0^2$, the standard $M_{\text{H}}$-${\sigma }_0$ form $(M_{\text{H}}\propto {\sigma }_0^4)$ is deduced from qualitative considerations. The projected bulge velocity dispersion to projected vortex velocity ratio, $\eta =({\sigma }_{\text{B}}{)}_{33}/\{[(v_{\text{V}}{)}_{\mathit{qq}}{]}^2+[({\sigma }_{\text{V}}{)}_{\mathit{qq}}{]}^2{\}}^{1/2}$, as a function of the fractional radius, $y_{\text{BV}}=R_{\text{B}}/R_{\text{V}}$, and the fractional masses, $m_{\text{BH}}=M_{\text{B}}/M_{\text{H}}$ and $m_{\text{BV}}=M_{\text{B}}/M_{\text{V}}$, is studied in the range of interest, $0?m_{\text{VH}}=M_{\text{V}}/M_{\text{H}}?5$ [Escala, A., 2006. ApJ, 648, L13] and $229?m_{\text{BH}}?795$ [Marconi, A., Hunt, L.H., 2003. ApJ 589, L21], consistent with observations. The related curves appear to be similar to Maxwell velocity distributions, which implies a fixed value of $\eta$ below the maximum corresponds to two different configurations: a compact bulge on the left of the maximum, and an extended bulge on the right. All curves lie very close one to the other on the left of the maximum, and parallel one to the other on the right. On the other hand, fixed $m_{\text{BH}}$ or $m_{\text{BV}}$, and $y_{\text{BV}}$, are found to imply more massive bulges passing from bottom to top along a vertical line on the $(\mathsf{O}{y}_{\text{BV}}\eta )$ plane, and vice versa. The model is applied to NGC 4374 and NGC 4486, taking the fractional mass,$m_{\text{BH}}$, and the fractional radius, $y_{\text{BV}}$, as unknowns, and the bulge mass is inferred from the knowledge of the hole mass, and compared with results from different methods. In presence of a massive vortex $(m_{\text{VH}}=5)$, the hole mass has to be reduced by a factor 2–3 with respect to the case of a massless vortex, to get the fit. Finally, the assumptions of homogeneous inner bulge and isotropic stress tensor are discussed.