Effect of core--mantle and tidal torques on Mercury's spin axis orientation

The rotational evolution of Mercury's mantle and its core under conservative and dissipative torques is important for understanding the planet's spin state. Dissipation results from tides and viscous, magnetic and topographic core--mantle interactions. The dissipative core--mantle torques take the system to an equilibrium state wherein both spins are fixed in the frame precessing with the orbit, and in which the mantle and core are differentially rotating. This equilibrium exhibits a mantle spin axis that is offset from the Cassini state by larger amounts for weaker core--mantle coupling for all three dissipative core--mantle coupling mechanisms, and the spin axis of the core is separated farther from that of the mantle, leading to larger differential rotation. The relatively strong core--mantle coupling necessary to bring the mantle spin axis to its observed position close to the Cassini state is not obtained by any of the three dissipative core--mantle coupling mechanisms. For a hydrostatic ellipsoidal core--mantle boundary, pressure coupling dominates the dissipative effects on the mantle and core positions, and dissipation together with pressure coupling brings the mantle spin solidly to the Cassini state. The core spin goes to a position displaced from that of the mantle by about 3.55 arcmin nearly in the plane containing the Cassini state. With the maximum viscosity considered of $\nu\sim 15.0\,{\rm cm^2/s}$ if the coupling is by the circulation through an Ekman boundary layer or $\nu\sim 8.75\times 10^5\,{\rm cm^2/s}$ for purely viscous coupling, the core spin lags the precessing Cassini plane by 23 arcsec, whereas the mantle spin lags by only 0.055 arcsec. Larger, non hydrostatic values of the CMB ellipticity also result in the mantle spin at the Cassini state, but the core spin is moved closer to the mantle spin.Comment: 35 pages, 7 figure

Relativistic Resonant Relations between Massive Black Hole Binary and Extreme Mass Ratio Inspiral

One component of a massive black hole binary (MBHB) might capture a small third body, and then a hierarchical, inclined triple system would be formed. With the post-Newtonian approximation including radiation reaction, we analyzed the evolution of the triple initially with small eccentricities. We found that an essentially new resonant relation could arise in the triple system. Here relativistic effects are crucial. Relativistic resonances, including the new one, stably work even for an outer MBHB of comparable masses, and significantly change the orbit of the inner small body.Comment: 9 pages, 5 figures, to appear in PR

Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets

(Abridged) A dynamical fit has placed the two planets about the star GJ 876 in coplanar orbits deep in 3 resonances at the 2:1 mean-motion commensurability with small libration amplitudes. The libration of both lowest order mean-motion resonance variables, theta_1 and theta_2, and the secular resonance variable, theta_3, about 0 deg. differs from the familiar geometry of the Io-Europa pair, where theta_2 and theta_3 librate about 180 deg. By considering a condition for stable simultaneous librations of theta_1 and theta_2, we show that the GJ 876 geometry results because of the large orbital eccentricities e_i, whereas the very small e_i in the Io-Europa system lead to the latter's geometry. Surprisingly, the GJ 876 resonance configuration remains stable for e_1 up to 0.86 and for amplitude of libration of theta_1 approaching 45 deg. with the current e_i. We find that inward migration of the outer planet of the GJ 876 system results in certain capture into the observed resonances if initially e_1 <0.06 and e_2<0.03 and the migration rate |(da_2/dt)/a_2| < 0.03(a_2/AU)^{-3/2} yr^{-1}. The bound on the migration rate is easily satisfied by migration due to planet-nebula interaction. If there is no eccentricity damping, eccentricity growth is rapid with continued migration within the resonance, with e_i exceeding the observed values after a further reduction in the semi-major axes a_i of only 7%. With eccentricity damping (de_i/dt)/e_i = -K|(da_i/dt)/a_i|, the e_i reach equilibrium values that remain constant for arbitrarily long migration within the resonances. The equilibrium e_i are close to the observed e_i for K=100 (K=10) if there is migration and damping of the outer planet only (of both planets). It is as yet unclear that planet-nebula interaction can produce the large value of K required to obtain the observed eccentricities.Comment: 23 pages, including 8 figures; uses AASTeX v5.0; minor additions; accepted for publication in Ap

Origin of Tidal Dissipation in Jupiter: II. the Value of Q

The process of tidal dissipation inside Jupiter is not yet understood. Its tidal quality factor ($Q$) is inferred to lie between $10^5$ and $10^6$. We examine effects of inertial-modes on tidal dissipation in a neutrally bouyant, core-less, uniformly rotating planet. The rate of dissipation caused by resonantly excited inertial-modes depends on the following three parameters: how well they are coupled to the tidal potential, how strongly they are dissipated (by the turbulent viscosity), and how densely distributed they are in frequency. We find that as a function of tidal frequency, the $Q$ value exhibits large fluctuations, with its maximum value set by the group of inertial-modes that have a typical offset from an exact resonance of order their turbulent damping rates. In our model, inertial-modes shed their tidally acquired energy very close to the surface within a narrow latitudinal zone (the 'singularity belt'), and the tidal luminosity escapes freely out of the planet. Strength of coupling between the tidal potential and inertial-modes is sensitive to the presence of density discontinuities inside Jupiter. In the case of a discreet density jump (as may be caused by the transition between metallic and molecular hydrogen), we find a time-averaged $Q \sim 10^7$. Even though it remains unclear whether tidal dissipation due to resonant inertial-modes is the correct answer to the problem, it is impressive that our simple treatment here already leads to three to five orders of magnitude stronger damping than that from the equilibrium tide. Moreover, our conclusions are not affected by the presence of a small solid core, a different prescription for the turbulent viscosity, or nonlinear mode coupling, but they depend critically on the static stability in the upper atmosphere of Jupiter.Comment: 27 pages, incl. 11 figures, ApJ in print, expanded discussions (nonlinearity, radiative envelope

Concentration of atomic hydrogen diffused into silicon in the temperature range 900–1300 °C

Boron-doped Czochralski silicon samples with [B]~1017 cm−3 have been heated at various temperatures in the range 800–1300 °C in an atmosphere of hydrogen and then quenched. The concentration of [H-B] pairs was measured by infrared localized vibrational mode spectroscopy. It was concluded that the solubility of atomic hydrogen is greater than [Hs] = 5.6 × 1018 exp( − 0.95 eV/kT)cm−3 at the temperatures investigated

Obliquity Tides on Hot Jupiters

Obliquity tides are a potentially important source of heat for extrasolar planets on close-in orbits. Although tidal dissipation will usually reduce the obliquity to zero, a nonzero obliquity can persist if the planet is in a Cassini state, a resonance between spin precession and orbital precession. Obliquity tides might be the cause of the anomalously large size of the transiting planet HD 209458b.Comment: To appear in ApJ Letters [9 pages, 2 figures

On the evolution of mean motion resonances through stochastic forcing: Fast and slow libration modes and the origin of HD128311

Aims. We clarify the response of extrasolar planetary systems in a 2:1 mean motion commensurability with masses ranging from the super Jovian range to the terrestrial range to stochastic forcing that could result from protoplanetary disk turbulence. The behaviour of the different libration modes for a wide range of system parameters and stochastic forcing magnitudes is investigated. The growth of libration amplitudes is parameterized as a function of the relevant physical parameters. The results are applied to provide an explanation of the configuration of the HD128311 system. Methods. We first develop an analytic model from first principles without making the assumption that both eccentricities are small. We also perform numerical N-body simulations with additional stochastic forcing terms to represent the effects of putative disk turbulence. Results. Systems are quickly destabilized by large magnitudes of stochastic forcing but some stability is imparted should systems undergo a net orbital migration. The slow mode, which mostly corresponds to motion of the angle between the apsidal lines of the two planets, is converted to circulation more readily than the fast mode which is associated with oscillations of the semi-major axes. This mode is also vulnerable to the attainment of small eccentricities which causes oscillations between periods of libration and circulation. Conclusions. Stochastic forcing due to disk turbulence may have played a role in shaping the configurations of observed systems in mean motion resonance. It naturally provides a mechanism for accounting for the HD128311 system.Comment: 15 pages, 8 figures, added discussion in h and k coordinates, recommended for publicatio

Site-Selective Spectroscopy And Crystal-Field Analysis For Nd3+ In Strontium Fluorovanadate

Site‐selective spectroscopy reveals that Nd3+ ions occupy more than 40 different crystal‐field environments in Sr5(VO4)3F. Preferential energy transfer to the site responsible for 1 μm lasing occurs but becomes less complete with increasing temperature. The 4I and 4F3/2 Stark levels of the lasing site have been determined and an analysis of the crystal field performed. From the crystal‐field fitting parameters Bkq, a calculated energy‐level spectrum is determined up to 17 500 cm−1 with a rms deviation from the available experimental levels of 6 cm−1