Semi-major Axis Ratio Research Articles

Elastic fracture mechanics analysis of thick-walled curved tubing with a semi-elliptical surface crack by the boundary element method

In this paper, the boundary element method is used to obtain the mode I stress intensity factors (SIFs) for a semi-elliptical crack in thick-walled curved tubing or elbow under internal pressure. A relatively wide range of geometric parameters for the tubing – the bend radius ratio, the cross-sectional radius ratio, and the angular extent of the circular bend – is considered. For each case of the curved tube geometry, a crack of semi-minor to semi-major axis ratio of 0.8, with depth varying from 20 per cent to 80 per cent of the wall thickness at the intrados is analysed. The computed values of the normalized SIFs are shown to be higher than those corresponding to a straight cylinder for the same relative crack depth and cross-sectional radius ratio. They are also found not to vary significantly along much of the crack periphery in the cases considered, increasing rapidly only as the free surface is approached. Furthermore, for a given relative crack size and location, they increase with decreasing bend radius ratio and decreasing cross-sectional radius ratio, but are, however, less sensitive to the angular extent of the elbow.

Transiting planets - light-curve analysis for eccentric orbits

Transiting planet light curves have historically been used predominantly for measuring the depth and hence ratio of the planet–star radii, p. Equations have previously been presented by Seager & Mallén-Ornelas for the analysis of the total and trough transit light-curve time to derive the ratio of semimajor axis to stellar radius, a/R*, in the case of circular orbits. Here, a new analytic model is proposed which operates for the more general case of an eccentric orbit. We aim to investigate three major effects our model predicts: (i) the degeneracy in transit light-curve solutions for eccentricity, e > 0; (ii) the asymmetry of the light curve and the resulting shift in the mid-transit time, TMID; (iii) the effect of eccentricity on the ingress and egress slopes. It is also shown that a system with changing eccentricity and inclination may produce a long-term transit time variation (LTTV). Furthermore, we use our model in a re-analysis of HD 209458b archived data by Richardson et al., where we include the confirmed non-zero eccentricity and derive a 24-μm planetary radius of RP= 1.275RJ± 0.082RJ (where RJ= 1 Jovian radius), which is ∼1 per cent larger than if we assume a circular orbit.

Prospects for the Habitability of OGLE-2006-BLG-109L

The extrasolar system OGLE-2006-BLG-109L is the first multiple-planet system to be discovered by gravitational microlensing (reported by Gaudi et al. in 2008); the two large planets that have been detected have mass ratios, semimajor axis ratios, and equilibrium temperatures that are similar to those of Jupiter and Saturn; the mass of the host star is only 0.5 M☉, and the system is more compact than our own solar system. We find that in the habitable zone of the host star, the two detected planets resonantly excite large orbital eccentricities on a putative Earth-mass planet, driving such a planet out of the habitable zone. We show that an additional inner planet of 0.3 M⊕ at 0.1 AU would suppress the eccentricity perturbation and greatly improve the prospects for habitability of the system. Thus, the planetary architecture of a potentially habitable OGLE-2006-BLG-109L planetary system—with two terrestrial planets and two Jovian planets—could bear very close resemblance to our own solar system.

An analysis of the transit times of TrES-1b

The presence of a second planet in a known, transiting-planet system will cause the time between transits to vary. These variations can be used to constrain the orbital elements and mass of the perturbing planet. We analyse the set of transit times of the TrES-1 system given in Charbonneau et al. (2005). We find no convincing evidence for a second planet in the TrES-1 system from that data. By further analysis, we constrain the mass that a perturbing planet could have as a function of the semi-major axis ratio of the two planets and the eccentricity of the perturbing planet. Near low-order, mean-motion resonances (within ~1% fractional deviation), we find that a secondary planet must generally have a mass comparable to or less than the mass of the Earth--showing that this data is the first to have sensitivity to sub Earth-mass planets. We compare the sensitivity of this technique to the mass of the perturbing planet with future, high-precision radial velocity measurements.

Analytical Approach to the Secular Behaviour of Exoplanetary Systems

We analyse the secular interactions of two coplanar planets which are not in mean motion resonances. The analysis is based on a high order (order 12) expansion of the perturbative potential in powers of the eccentricities. The model depends on only two parameters (the ratio of semi-major axis and the mass ratio of the planets) and can be reduced to a one degree of freedom system, allowing for an exhaustive parametric analysis. Following Pauwels [Pauwels T.: 1983, Celet. Mech. & Dyn. Astro. 30, 229–247] we map the phase space on a sphere, avoiding in this way the artificial singularities introduced by other mappings. We show that the 12 order expansion is able to describe correctly most of the exosolar planetary systems discovered so far, even if the eccentricities of these planets are considerably larger than the eccentricities of our own solar system. The expansion is even able to reproduce, at moderate eccentricities, the secular resonances discovered numerically by Michtchenko and Malhotra [Michtchenko, T. A. and Malhotra, R.: 2004, Icarus 168, 237–248] at moderate to large eccentricities.

Triaxial haloes, intrinsic alignments and the dark matter power spectrum

We develop the halo model of large-scale structure to include triaxial dark matter haloes and their intrinsic alignments. As a direct application we derive general expressions for the two-point correlation function and the power spectrum. We then focus on the power spectrum and numerically solve the general expressions for two different models of the triaxial profiles. The first is a toy model that allows us to isolate the dependence of clustering on halo shape alone and the second is the more realistic profile model of Jing & Suto. In both cases, we find that the effect of triaxiality is manifest as a suppression of power at the level of ∼5 per cent on scales k∼ 1–10hMpc−1, which in real space corresponds to the virial radii of clusters. When considered by mass, we find that for the first model the effects are again apparent as a suppression of power and that they are more significant for the high-mass haloes. For the Jing & Suto model, we find a suppression of power on large scales followed by a sharp amplification on small scales at the level of ∼10–15 per cent. Interestingly, when averaged over the entire mass function this amplification effect is suppressed. We also find for the one-halo term on scales k < 10hMpc−1 that the power is dominated by ellipsoidal haloes with semi-minor to semi-major axis ratios a/c < 0.7. One of the important features of our formalism is that it allows for the self-consistent inclusion of the intrinsic alignments of haloes. The alignments are specified through the correlation function of halo seeds. We develop a useful toy model for this and then make estimates of the alignment contribution to the power spectrum. Further, through consideration of the (artificial) case where all haloes are perfectly aligned, we calculate the maximum possible contribution to the clustering. We find the hard limit of <10 per cent. Subject to further scrutiny, the proposed toy model may serve as a means for linking the actual observed intrinsic alignments of galaxies to physical quantities of interest.

Stability of Minor-Body Orbits in Systems with Two Giant Planets

We have performed a large ensemble of long-term numerical integrations to study the stability of the orbits of minor bodies in systems containing a solar-mass star and two giant planets. Given the large parameter space involved, we have focused on systems in which the inner planet has the mass of Jupiter (MJ) and the outer planet has a mass equal to either MJ or MJ, and where we use an initially dynamically cold (e = 0) minor-body population. We investigated the effects of the planetary semimajor-axis ratio, eccentricity, and inclination on the stability of orbits distributed throughout the system. We show that the behavior of the particles varies from region to region as the result of a complex interplay of the two major types of resonances: resonances associated with commensurabilities of orbital frequencies, and resonances associated with commensurabilities of orbital precession frequencies. In the region inward of the inner planet, mean motion resonances produce instabilities and secular resonances induce high eccentricities in the particles. Between the planets, the mean motion resonances are dominant and generally induce instability. Beyond the outer planet, secular and mean motion resonances overlap and produce wide-scale instability. In this last region, many stable particles are associated with mean motion resonances in various kinds of protective mechanisms. We also show that increased planetary eccentricity generally results in increased instability and that high initial inclination of both planet and test particles greatly changes the final structure of the system. Overall, our results show trends that make it possible to predict the general features of the minor-body distribution for a given set of orbital elements of the planets. We demonstrate this with two examples drawn from the set of observed extrasolar planetary systems.

Secular dynamics of the three-body problem: application to the υ Andromedae planetary system

The discovery of extra-solar planetary systems with multiple planets in highly eccentric orbits (∼0.1–0.6), in contrast with our own Solar System, makes classical secular perturbation analysis very limited. In this paper, we use a semi-numerical approach to study the secular behavior of a system composed of a central star and two massive planets in co-planar orbits. We show that the secular dynamics of this system can be described using only two parameters, the ratios of the semi-major axes and the planetary masses. The main dynamical features of the system are presented in geometrical pictures that allows us to investigate a large domain of the phase space of this three-body problem without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The topology of the phase space is also investigated in detail by means of spectral map techniques, which allow us to detect the separatrix of a non-linear secular apsidal resonance. Finally, the qualitative study is supplemented by direct numerical integrations. Three different regimes of secular motion with respect to the secular angle Δ ϖ are possible: they are circulation, oscillation (around 0° and 180°), and high eccentricity libration in a non-linear secular resonance. The first two regimes are a continuous extension of the classical linear secular perturbation theory; the last is a new feature, hitherto unknown, in the secular dynamics of the three-body problem. We apply the analysis to the case of the two outer planets in the υ Andromedae system, and obtain its periodic and ordinary orbits, the general structure of its secular phase space, and the boundaries of its secular stability; we find that this system is secularly stable over a large domain of eccentricities. Applying this analysis to a wide range of planetary mass and semi-major axis ratios (centered about the υ Andromedae parameters), we find that apsidal oscillation dominates the secular phase space of the three-body coplanar system, and that the non-linear secular resonance is also a common feature.

Occurrence and Stability of Apsidal Resonance in Multiple Planetary Systems

With the help of the Laplace-Lagrange solution of the secular perturbation theory in a double-planet system, we study the occurrence and the stability of apsidal secular resonance between the two planets. The explicit criteria for predicting whether two planets are in apsidal resonance is derived, which shows that the occurrence of the apsidal resonance depends only on the mass ratio (m1/m2), semimajor axis ratio (a1/a2), initial eccentricity ratio (e10/e20), and the initial relative apsidal longitude (20 - 10) between the two planets. The probability of two planets falling in apsidal resonance is given in the initial element space. We verify the criteria with numerical integrations for the HD 12661 system and find they give good predictions except at the boundary of the criteria or when the planet eccentricities are too large. The nonlinear stability of the two planets in HD 12661 system are studied by calculating the Lyapunov exponents of their orbits in a general three-body model. We find that two planets in large-eccentricity orbits could be stable only when they are in aligned apsidal resonance. When the planets are migrated under the planet-disk interactions, for more than half of the studied cases, the configurations of the apsidal resonances are preserved. We find the two planets of the HD 12661 system could be in aligned resonance and thus more stable, provided they have Ω2 - Ω1 ≈ 180°. The applications of the criteria to the other multiple planetary systems are discussed.

Eccentricity Evolution of Extrasolar Multiple Planetary Systems Due to the Depletion of Nascent Protostellar Disks

Most extrasolar planets are observed to have eccentricities much larger than those in the solar system. Some of these planets have sibling planets, with comparable masses, orbiting around the same host stars. In these multiple planetary systems, eccentricity is modulated by the planets' mutual secular interaction as a consequence of angular momentum exchange between them. For mature planets, the eigenfrequencies of this modulation are determined by their mass and semi-major axis ratios. But, prior to the disk depletion, self gravity of the planets' nascent disks dominates the precession eigenfrequencies. We examine here the initial evolution of young planets' eccentricity due to the apsidal libration or circulation induced by both the secular interaction between them and the self gravity of their nascent disks. We show that as the latter effect declines adiabatically with disk depletion, the modulation amplitude of the planets' relative phase of periapse is approximately invariant despite the time-asymmetrical exchange of angular momentum between planets. However, as the young planets' orbits pass through a state of secular resonance, their mean eccentricities undergo systematic quantitative changes. For applications, we analyze the eccentricity evolution of planets around Upsilon Andromedae and HD168443 during the epoch of protostellar disk depletion. We find that the disk depletion can change the planets' eccentricity ratio. However, the relatively large amplitude of the planets' eccentricity cannot be excited if all the planets had small initial eccentricities.

Tidal evolution of the Uranian satellites: III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and Ariel-Umbriel 2:1 mean-motion commensurabilities

Tidal evolution has significantly affected the history of the Uranian satellite system. The presence of large chaotic zones at past mean-motion commensurabilities among the Uranian satellites has resulted in significant changes in the orbital elements of some of the satellites, while allowing them to escape from the commensurabilities and evolve to their present nonresonant configuration. Miranda and Umbriel have probably passed through the 3:1 commensurability, resulting in Miranda's current anomalously high inclination, and constraining the Q of Uranus to be less than 39,000 (W. C. Tittemore and J. Wisdom 1989, Icarus 78, 63–89) . The orbits of both satellites become chaotic during evolution through the eccentricity resonances associated with the 3:1 commensurability. During this phase of evolution, the orbital eccentricity of Miranda can be driven up to a value of about 0.05. Miranda can then escape from the 3:1 commensurability with a relatively large orbital eccentricity, which can damp to the current value in the time since resonance passage. Tidal friction may have heated the interior of Miranda to a temperature near the eutectic melting point of NH 3 · H 2O, but most likely did not result in the melting of significant quantities of water ice. Miranda and Ariel passed through the 5:3 mean-motion commensurability if the Q of Uranus is less than about 12,000. During evolution through this commensurability, the semimajor axis ratio ( a M/ a A) decreased. As the orbits enter a large chaotic zone associated with this commensurability, both the eccentricity and inclination of Miranda's orbit jump to values up to six times higher than the values approaching the resonance. Upon escaping from the resonance, the orbit of Miranda may have retained moderately high eccentricity and inclination, or e M and i M may have decreased back to values comparable to those approaching the resonance. If the Q of Uranus is smaller than about 11,000, Ariel and Umbriel would have encountered the 2:1 mean-motion commensurability. However, capture into this resonance is very likely if the eccentricities of the orbits approaching the resonance were comparable to the current values: the probability of escape would not have been significant unless the initial eccentricities were of order 0.03 or larger. It is therefore unlikely that these satellites encountered this resonance, constraining the Q of Uranus to be greater than 11,000. The specific dissipation function of Uranus is therefore well constrained: 11,000 < Q < 39,000.

Wind‐erosion Direction Factors as Influenced by Field Shape and Wind Preponderance

AbstractThis investigation expands and improves the procedure for determining median travel distance of wind in traversing a field, information used in solving the wind‐erosion equation. Wind‐erosion roses were simulated by the equation of an ellipse in polar coordinates. The ratio of semimajor axis to semiminor axis was varied to give preponderance values from 1.0 to 4.0. The axis was rotated to simulate field orientation from 0 to 90° relative to prevailing winderosion direction. Length/width ratio for rectangular fields was varied from 1 to 10. The wind‐erosion direction factor, a number that when multiplied by field width gives median travel distance, was calculated for many combinations of variables. When preponderance was 1.0, the wind‐erosion direction factor was 1.03, 1.42, 1.48, and 1.48 for rectangular fields with length/width ratios of 1, 2, 4, and 10, respectively; the factor was not influenced by field orientation. As preponderance increased, the wind‐erosion direction factor approached unity for small angles of deviation and approached the length/width ratio for large angles of deviation. For circular fields surrounded by a nonerodible surface, the wind‐erosion direction factor was 0.91, regardless of wind direction and preponderance.

Spectral Domain Analysis of Elliptic Microstrip Disk Resonators

The elliptic microstrip disk resonator in an open microstrip configuration is analyzed using the spectral domain technique under quasi-static approximation. The normalized value of capacitance is presented as a function of the ratio of substrate thickness and semimajor axis of the elliptic disk resonator for different values of eccentricity. The fringing field effects associated with the structure are quantitatively assessed in terms of effective normalized semimajor axis. The resonator frequencies computed incorporating this effect are in good agreement with the experimental values reported in the literature. Results for various dielectric constants are also included.