118852244
Exponential growth of eccentricity in secular theory
The Kozai mechanism for exponentially exciting eccentricity of a Keplerian orbit by a distant perturber is extended to a general perturbing potential. In particular, the case of an axisymmetric potential is solved analytically. The analysis is applied to orbits around an oblate central object with a distant perturber. If the equatorial plane of the central object is aligned with the orbit of the distant perturber (axisymmetric potential), a single instability zone, in which eccentricity grows exponentially, is found between two critical inclinations; if misaligned (non-axisymmetric potential), a rich set of critical inclinations separating stable and unstable zones is obtained (Vashkoviak 1974). The analysis is also applied to a general quadratic potential. Similarly, for non-axisymmetric cases, multiple stability and instability zones are obtained. Here eccentricity can reach very high values in the instability zones even when the potential's deviation from axisymmetry is small.
118316787
An exact integral relation between the Ni56 mass and the bolometric light curve of a type Ia supernova
An exact relation between the Ni56 mass and the bolometric light curve of a type Ia supernova can be derived as follows, using the following excellent approximations: 1. the emission is powered solely by Ni56-> Co56 ->Fe56; 2. each mass element propagates at a non-relativistic velocity which is constant in time (free coasting); and 3. the internal energy is dominated by radiation. Under these approximations, the energy E(t) carried by radiation in the ejecta satisfies: dE/dt=-E(t)/t-L(t)+Q(t), where Q(t) is the deposition of energy by the decay which is precisely known and L(t) is the bolometric luminosity. By multiplying this equation by time and integrating over time we find: E(t)*t=\int_0^t Q(t')t'dt' -\int_0^t L(t')t'dt'. At late time, t>> t_peak, the energy inside the ejecta decreases rapidly due to its escape, and thus we have \int_0^t Q(t')t'dt'=\int_0^t L(t')t'dt'. This relation is correct regardless of the opacities, density distribution or Ni56 deposition distribution in the ejecta and is very different from "Arnett's rule", L_peak ~ Q(t_peak). By comparing \int_0^t Q(t')t'dt' with \int_0^t L(t')t'dt' at t~40 day after the explosion, the mass of Ni56 can be found directly from UV, optical and infrared observations with modest corrections due to the unobserved gamma-rays and due to the small residual energy in the ejecta, E(t)*t>0.
119233296
THE STATISTICS OF MULTI-PLANET SYSTEMS
We describe statistical methods for measuring the exoplanet multiplicity function (the fraction of host stars containing a given number of planets) and inclination distribution from transit and radial-velocity (RV) surveys. The analysis is based on the approximation of separability-that the distribution of planetary parameters in an n-planet system is the product of identical 1-planet distributions. We review the evidence that separability is a valid approximation for exoplanets and conclude that it captures many, but not all, of the known characteristics of multi-planet systems. We show how to relate the observable multiplicity function in surveys with similar host-star populations but different sensitivities. We also show how to correct for geometrical selection effects to derive the multiplicity function from transit surveys if the distribution of relative inclinations is known. Applying these tools to the Kepler transit survey and to RV surveys, we find that (1) the Kepler data alone do not constrain the mean inclination of multi-planet systems; even spherical distributions are allowed by the data but only if a small fraction of host stars contain large planet populations ({approx}> 30); (2) comparing the Kepler and RV surveys shows that the mean inclination of multi-planet systems is less than 5 Degree-Sign ;more » and (3) the multiplicity function of the Kepler planets is not well determined by the present data.« less
20954769
MOA-2009-BLG-387Lb: A massive planet orbiting an M dwarf
Aims. We report the discovery of a planet with a high planet-to-star mass ratio in the microlensing event MOA-2009-BLG-387, which exhibited pronounced deviations over a 12-day interval, one of the longest for any planetary event. The host is an M dwarf, with a mass in the range 0.07 M_⊙ < M_(host) < 0.49 M_⊙ at 90% confidence. The planet-star mass ratio q = 0.0132 ± 0.003 has been measured extremely well, so at the best-estimated host mass, the planet mass is m_p = 2.6 Jupiter masses for the median host mass, M = 0.19 M_⊙. Methods. The host mass is determined from two “higher order” microlensing parameters. One of these, the angular Einstein radius θ_E = 0.31 ± 0.03 mas has been accurately measured, but the other (the microlens parallax πE, which is due to the Earth’s orbital motion) is highly degenerate with the orbital motion of the planet. We statistically resolve the degeneracy between Earth and planet orbital effects by imposing priors from a Galactic model that specifies the positions and velocities of lenses and sources and a Kepler model of orbits. Results. The 90% confidence intervals for the distance, semi-major axis, and period of the planet are 3.5 kpc < D_L < 7.9 kpc, 1.1 AU < a < 2.7 AU, and 3.8 yr < P < 7.6 yr, respectively.
119112522
Type Ia supernovae with bimodal explosions are common – possible smoking gun for direct collisions of white dwarfs
We discover clear doubly-peaked line profiles in 3 out of ~20 type Ia supernovae (SNe Ia) with high-quality nebular-phase spectra. The profiles are consistently present in three well-separated Co/Fe emission features. The two peaks are respectively blue-shifted and red-shifted relative to the host galaxies and are separated by ~5000 km/s. The doubly-peaked profiles directly reflect a bi-modal velocity distribution of the radioactive Ni56 in the ejecta that powers the emission of these SNe. Due to their random orientations, only a fraction of SNe with intrinsically bi-modal velocity distributions will appear as doubly-peaked spectra. Therefore SNe with intrinsic bi-modality are likely common, especially among the SNe in the low-luminosity part on the Philips relation (\Delta m15(B) >~ 1.3; ~40% of all SNe Ia). Such bi-modality is naturally expected from direct collisions of white dwarfs (WDs) due to the detonation of both WDs and is demonstrated in a 3D 0.64 M_Sun-0.64 M_Sun WD collision simulation. In the future, with a large sample of nebular spectra and a comprehensive set of numerical simulations, the collision model can be unambiguously tested as the primary channel for type Ia SNe, and the distribution of nebular line profiles will either be a smoking gun or rule it out.
119257545
Head-on Collisions of White Dwarfs in Triple Systems Could Explain Type Ia Supernovae
Type Ia supernovae (SNe Ia), thermonuclear explosions of carbon-oxygen white dwarfs (CO-WDs), are currently the best cosmological "standard candles," but the triggering mechanism of the explosion is unknown. It was recently shown that the rate of head-on collisions of typical field CO-WDs in triple systems may be comparable to the SNe Ia rate. Here we provide evidence supporting a scenario in which the majority of SNe Ia are the result of such head-on collisions of CO-WDs. In this case, the nuclear detonation is due to a well understood shock ignition, devoid of commonly introduced free parameters such as the deflagration velocity or transition to detonation criteria. By using two-dimensional hydrodynamical simulations with a fully resolved ignition process, we show that zero-impact-parameter collisions of typical CO-WDs with masses 0.5-1 M ☉ result in explosions that synthesize 56Ni masses in the range of ~0.1-1 M ☉, spanning the wide distribution of yields observed for the majority of SNe Ia. All collision models yield the same late-time ( 60 days since explosion) bolometric light curve when normalized by 56Ni masses (to better than 30%), in agreement with observations. The calculated widths of the 56Ni-mass-weighted line-of-sight velocity distributions are correlated with the calculated 56Ni yield, agreeing with the observed correlation. The strong correlation, shown here for the first time, between 56Ni yield and total mass of the colliding CO-WDs (insensitive to their mass ratio), is suggestive as the source for the continuous distribution of observed SN Ia features, possibly including the Philips relation.
120777661
ANALYTICAL APPROXIMATIONS TO THE SCHRÖDINGER EQUATION FOR A SECOND PÖSCHL–TELLER-LIKE POTENTIAL WITH CENTRIFUGAL TERM
The bound state solutions of the Schrodinger equation for a second Poschl–Teller-like potential with the centrifugal term are obtained approximately. It is found that the solutions can be expressed in terms of the hypergeometric functions 2F1(a, b; c; z). To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other method for short-range potential. Two special cases for l = 0 and V1 = V2 are also studied briefly.
119637956
GENERALIZED HYPERVIRIAL AND RECURRENCE RELATIONS FOR RADIAL MATRIX ELEMENTS IN ARBITRARY DIMENSIONS
In this letter, based on Hamiltonian identity we propose a generalized expression of the second hypervirial for an arbitrary central potential wave functions in higher dimensions D and then present a useful general recurrence relation among the off-diagonal multipolar matrix elements. We show that this new proposed recurrence formula is very powerful in deriving the recurrence relations among the hydrogenic matrix elements. Some important recurrence relations can be simply obtained. It is interesting to find that the selection rule is independent of the central potential V(r) for f = rk with k = 0, 2 and that the diagonal matrix elements of V′(r) are independent of the system eigenvalues for k = 1. The application of the general recurrence relation to the isotropic harmonic oscillator case is discussed briefly.
120445308
On the Dirac Equation with a Coulomb Potential in D + 1 Dimensions
The Dirac equation is investigated in D + 1 dimensions. The exact solutions of the Dirac equation with the Coulomb potential are analytically obtained by studying the Tricomi equations. The eigenvalues and the corresponding fine structure are also discussed.
122483032
Wave Equations in Higher Dimensions
Part I (Introduction).- Part II (Theory). - 2. Special orthogonal groups (Introduction Abstract groups Orthogonal group SO(n) Tensor representations of the SO(n) \Gamma matrix groups Spinor representations of the SO(n) Concluding remarks).- 3. Rotational symmetry and Schrodinger equation in N-dimensional space (Introduction Rotation operator Orbital angular momentum operators The linear momentum operators Radial momentum operator Spherical harmonic polynomials Schrodinger equation for a two-body system Concluding remarks).- 4. Dirac equation in higher dimensions (Introduction Dirac equation in N+1 dimensions The radial equation Application to hydrogen atom Concluding remarks).- 5. Klein-Gordon equation in higher dimensions (Introduction The Radial equation Application to hydrogen atom Concluding remarks).- Part III (Application in Non-relativistic Quantum Mechanics).- 6. Harmonic oscillator (Introduction Exact solutions of harmonic oscillator Recurrence relations for the radioal function Realization of dynamic group SU(1, 1) Generalization to pseudoharmonic ooscillator Position and momentum information entropy Conclusions).- 7. Coulomb potential (Introduction Exact solution Shift operators Mapping between Coulumb and harmonic oscillator radial functions Realization of dynamic of dynamic group SU (1, 1) Generalization to Kratzer potential Concluding remarks).- 8. Wave function ansatz method (Introduction Sextic potential Singular one-fraction power potential Mixture potential Non-polynomial potential Screened Coulomb potential Morse potential Conclusions).- 9. Levinson theorem for Schrodinger equation (Introduction Scattering states and phase shifts Bound states Sturm--Liouville theorem Levinson theorem Discussions Conclusions).- 10. Generalized hypervirial theorem for Schrodinger equation (Introduction Generalized Blanchard's and Kramers' recurrence relations Applications to central potentials Conclusions).- 11. Exact quantization rule and Langer modification (Introduction WKB approximation Exact quantization rule Application to trigonometric Rosen-Morse potential Proper quantization rule Illustrations of proper quantization rule Langer modification in D dimensions Calculations of logarithmic derivatives of wavefunction Conclusions).- 12. Schrodinger equation with position-dependent mass (Introduction Formalism Applications to harmonic oscillator and Coulomb potential Conclusions).- Part IV (Application in Relativistic Quantum Mechanics).- 13. Dirac equation with Coulomb potential (Introduction Exact solutions of hydrogen-like atoms Analysis of eigenvalues Generalization to the Dirac equation with Coulomb potential plus scalar potential Concluding remarks).- 14. Klein-Gordon equation with Coulomb potential (Introduction Eigenfunctions and eigenvalues Analysis of eigenvalues Generalization: Klein-Gordon equation with Coulomb plus scalar potential Comparison theorem Conclusions).- 15. Levinson theorem for Dirac equation (Introduction Generalization Sturm-Liouville theorem Number of bound states Relativistic Levinson theorem Discussions Friedel Theorem Comparison theorem Conclusions).- 16. Generalized hypervirial theorem for Dirac equation (Introduction Relativistic recurrence relation Diagonal case Conclusions).- 17. Kaluza-Klein theory (Introduction (4+D) -dimensional Kaluza-Klein theories Paritcle spectrum of Kaluza-Klein theories for ferminions Warped extra dimensions Conclusions).- PART V (Conclusions and Outlooks).- 18. Conclusions and outlooks.- Appendices.- References.- Index.
117170107
A New Approach to the Relativistic Schrödinger Equation with Central Potential: Ansatz Method
Applying an ansatz to the eigenfunction, we obtain the exact closed-form solutions of the relativistic Schrödinger equation with the potential V(r) = −a/r + b/r1/2 both in three dimensions and in two dimensions. The restrictions on the parameters of the given potential and the angular momentum quantum number are also presented.