Stability of resonant planetary orbits in binary stars

This paper contains a numerical study of the stability of resonant orbits in a planetary system consisting of two planets, moving under the gravitational attraction of a binary star. Its results are expected to provide us with useful information about real planetary systems and, at the same time, about periodic motions in the general four-body problem (G4) because the above system is a special case of G4 where two bodies have much larger masses than the masses of the other two (planets). The numerical results show that the main mechanism which generates instability is the destruction of the Jacobi integrals of the massless planets when their masses become nonzero and that resonances in the motion of planets do not imply, in general, instability. Considerable intervals of stable resonant orbits have been found. The above quantitative results are in agreement with the existing qualitative predictions     

Stable planetary orbits around one component in nearby binary stars. II

Numerical simulations are made within the framework of the plane restricted three-body problem, in order to find out if stable orbits for planets around one of the two components in double stars can exist. For any given set of initial parameters (the mass ratio of the two stars and the eccentricity of their orbit around each other), the phase-space of initial positions and velocities is systematically explored.In previous works, systematic exploration of the circular model as well as studies of more realistic (elliptic) cases such as Sun-Jupiter and the nearby Centauri and Sirius systems, large stable planetary orbits were found to exist around both components of the binary, up to distances from each star of the order or more than half the binary's periastron separation.The first results presented here for the Coronae Borealis system confirm the previous studies.     

On the effect of the eccentricity of a planetary orbit on the stability of satellite orbits

The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic orbits, bifurcating from the Lagrangian points L₁, L₂ of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these exponents are positive, indicating the existence of chaotic motions     

Stability of axial orbits in analytical galactic potentials

We study the stability of axial orbits in analytical galactic potentials as a function of the energy of the orbit and the ellipticity of the potential. The problem is solved by an analytical method, the validity of which is not limited to small amplitudes. The lines of neutral stability divide the parameter space in regions corresponding to different organizations of the main families of orbits in the symmetry planes.     

Vertical stability of periodic solutions around the triangular equilibrium points

The vertical stability character of the families of short and long period solutions around the triangular equilibrium points of the restricted three-body problem is examined. For three values of the mass parameter less than equal to the critical value of Routh (μ _R ) i.e. for μ = 0.000953875 (Sun-Jupiter), μ = 0.01215 (Earth-Moon) and μ = μ _R = 0.038521, it is found that all such solutions are vertically stable. For μ > (μ _R ) vertical stability is studied for a number of ‘limiting’ orbits extended to μ = 0.45. The last limiting orbit computed by Deprit for μ = 0.044 is continued to a family of periodic orbits into which the well known families of long and short period solutions merge. The stability characteristics of this family are also studied.     

[WC] central stars of planetary nebula and the born-again phenomenon

[WC] central stars of planetary nebulae are members of the larger class of hydrogen-deficient central stars. The whole class constitutes about20% of all spectroscopically-known central stars. Observational connections between [WC] central stars and the born-again phenomenon show that at least a fraction of the [WC] stars can be createdthrough this scenario. However, it is unlikely that the class as a wholeevolved through this channel.In this paper the arguments against a born-again origin for the whole class of [WC] central stars of planetary nebula are outlined. It is suggested that the roleof the H-deficient weak emission lines stars might be crucial in explaining the origin of [WC]stars. It is also demonstrated how difficult it isto pin down the exact stellar parameters of these objects (which help toposition them on the HR diagram). This is due to the largely unknown distancesand to the fact that small changesin the model assumptions can have large repercussions on the derived parameters.This difficultyhampers our efforts to determine the true evolutionary position of individual [WC] central stars, as well as their relationship to one another, andtherefore pin down their origin.     

Distribution of periodic orbits and the homoclinic tangle

We study the distribution of regular and irregular periodic orbits on a Poincaré surface of section of a simple Hamiltonian system of 2 degrees of freedom. We explain the appearance of many lines of periodic orbits that form Farey trees. There are also lines that are very close to the asymptotic curves of the unstable periodic orbits. Some regular orbits, sometimes stable, are found inside the homoclinic tangle. We explain this phenomenon, which shows that the homoclinic tangle does not cover the whole area around an unstable orbit, but has gaps. Inside the lobes only irregular orbits appear, and some of them are stable. We conjecture that the opposite is also true, i.e. all irregular orbits are inside lobes.     

Continuity and stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron. 78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given numerically, and some examples are given of nearby stable orbits which bifurcate from these families.     

Stability of periodic solutions for Hill’s averaged problem with allowance for planetary oblateness

We analyze the stability of periodic solutions for Hill’s double-averaged problem by taking into account a central planet’s oblateness. They are generated by steady-state solutions that are stable in the linear approximation. By numerically calculating the monodromy matrix of variational equations, we plot its trace against the integral of the problem—an averaged perturbing function, for two model systems, [(Sun + Moon)-Earth-satellite] and (Sun-Uranus-satellite). We roughly estimate the ranges of values for the parameters of satellite orbits corresponding to periodic solutions of the evolutionary system that are stable in the linear approximation.     

The Existence and Stability of Families of Displacement Two-Body Orbits

The two-body problem is considered with an additional thrust induced acceleration. Stationary solutions are obtained to this problem in a rotating frame of reference which generates families of displaced circular orbits when viewed from an internal frame of reference. The existence and stability of these orbits is considered along with applications such as in-situ observations of Saturn's ring system and spacecraft proximity operations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

Moonlet induced wakes in planetary rings: Analytical model including eccentric orbits of moon and ring particles

Saturn’s rings host two known moons, Pan and Daphnis, which are massive enough to clear circumferential gaps in the ring around their orbits. Both moons create wake patterns at the gap edges by gravitational deflection of the ring material (Cuzzi, J.N., Scargle, J.D. [1985]. Astrophys. J. 292, 276-290; Showalter, M.R., Cuzzi, J.N., Marouf, E.A., Esposito, L.W. [1986]. Icarus 66, 297-323). New Cassini observations revealed that these wavy edges deviate from the sinusoidal waveform, which one would expect from a theory that assumes a circular orbit of the perturbing moon and neglects particle interactions. Resonant perturbations of the edges by moons outside the ring system, as well as an eccentric orbit of the embedded moon, may partly explain this behavior (Porco, C.C., and 34 colleagues [2005]. Science 307, 1226-1236; Tiscareno, M.S., Burns, J.A., Hedman, M.M., Spitale, J.N., Porco, C.C., Murray, C.D., and the Cassini Imaging team [2005]. Bull. Am. Astron. Soc. 37, 767; Weiss, J.W., Porco, C.C., Tiscareno, M.S., Burns, J.A., Dones, L. [2005]. Bull. Am. Astron. Soc. 37, 767; Weiss, J.W., Porco, C.C., Tiscareno, M.S. [2009]. Astron. J. 138, 272-286). Here we present an extended non-collisional streamline model which accounts for both effects. We describe the resulting variations of the density structure and the modification of the nonlinearity parameter q. Furthermore, an estimate is given for the applicability of the model. We use the streamwire model introduced by Stewart (Stewart, G.R. [1991]. Icarus 94, 436-450) to plot the perturbed ring density at the gap edges.We apply our model to the Keeler gap edges undulated by Daphnis and to a faint ringlet in the Encke gap close to the orbit of Pan. The modulations of the latter ringlet, induced by the perturbations of Pan (Burns, J.A., Hedman, M.M., Tiscareno, M.S., Nicholson, P.D., Streetman, B.J., Colwell, J.E., Showalter, M.R., Murray, C.D., Cuzzi, J.N., Porco, C.C., and the Cassini ISS team [2005]. Bull. Am. Astron. Soc. 37, 766), can be well described by our analytical model. Our analysis yields a Hill radius of Pan of 17.5 km, which is 9% smaller than the value presented by Porco (Porco, C.C., and 34 colleagues [2005]. Science 307, 1226-1236), but fits well to the radial semi-axis of Pan of 17.4 km. This supports the idea that Pan has filled its Hill sphere with accreted material (Porco, C.C., Thomas, P.C., Weiss, J.W., Richardson, D.C. [2007]. Science 318, 1602-1607). A numerical solution of a streamline is used to estimate the parameters of the Daphnis-Keeler gap system, since the close proximity of the gap edge to the moon induces strong perturbations, not allowing an application of the analytic streamline model. We obtain a Hill radius of 5.1 km for Daphnis, an inner edge variation of 8 km, and an eccentricity for Daphnis of 1.5 × 10⁻⁵. The latter two quantities deviate by a factor of two from values gained by direct observations (Jacobson, R.A., Spitale, J., Porco, C.C., Beurle, K., Cooper, N.J., Evans, M.W., Murray, C.D. [2008]. Astron. J. 135, 261-263; Tiscareno, M.S., Burns, J.A., Hedman, M.M., Spitale, J.N., Porco, C.C., Murray, C.D., and the Cassini Imaging team [2005]. Bull. Am. Astron. Soc. 37, 767), which might be attributed to the neglect of particle interactions and vertical motion in our model.     

Secular dynamics of S-type planetary orbits in binary star systems: applicability domains of first- and second-order theories

We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturbations are sufficiently small. The second-order model, although more complex, has a larger range of validity and must be taken into account for dynamical studies of some real exoplanetary systems such as \(\gamma \) Cephei and HD 41004A. However, in some extreme cases, neither of these analytical models yields quantitatively correct results, requiring either higher-order theories or direct numerical simulations. Finally, we determine the limits of applicability of each analytical model in the parameter space of the system, giving an important visual aid to decode which secular theory should be adopted for any given planetary system in a close binary.     

Stability Regions and Quasi-Periodic Motion in the Vicinity of Triangular Equilibrium Points

The regions of quasi-periodic motion around non-symmetric periodic orbits in the vicinity of the triangular equilibrium points are studied numerically. First, for a value of the mass parameter less than Routh's critical value, the stability regions determined by quasi-periodic motion are examined around the existing families of short (L^s ₄) and long (L^l ₄) period solutions. Then, for two values of μ greater than the Routh value, the unified family L^sl ₄, to which, in these cases, L^s ₄ and L^l ₄ merge, is considered. It is found that such regions surround in general the linearly stable segments of the corresponding families and become smaller as the mass ratio increases. This revised version was published online in July 2006 with corrections to the Cover Date.     

The full set of gas giant structures: I: On the origin of planetary masses and the planetary initial mass function

Context

Current planet search programs are detecting extrasolar planets at a rate of 60 planets per year. These planets show more diverse properties than was expected.

Aims

We try to get an overview of possible gas giant (proto-) planets for a full range of orbital periods and stellar masses. This allows the prediction of the full range of possible planetary properties which might be discovered in the near future.

Methods

We calculate the purely hydrostatic structure of the envelopes of proto-planets that are embedded in protoplanetary disks for all conceivable locations: combinations of different planetesimal accretion rates, host star masses, and orbital separations. At each location all hydrostatic equilibrium solutions to the planetary structure equations are determined by variation of core mass and pressure over many orders of magnitude. For each location we analyze the distribution of planetary masses.

Results

We get a wide spectrum of core-envelope structures. However, practically all calculated proto-planets are in the planetary mass range. Furthermore, the planet masses show a characteristic bimodal, sometimes trimodal, distribution. For the first time, we identify three physical processes that are responsible for the three characteristic planet masses: self-gravity in the Hill sphere, compact objects, and a region of very low adiabatic pressure gradient in the hydrogen equation of state. Using these processes, we can explain the dependence of the characteristic masses on the planet’s location: orbital period, host star mass, and planetesimal accretion rate (luminosity). The characteristic mass caused by the self-gravity effect at close proximity to the host star is typically one Neptune mass, thus producing the so-called hot Neptunes.

Conclusions

Our results suggest that hot Jupiters with orbital period less than 64 days (the exact location of the boundary depends on stellar type and accretion rate) have quite distinct properties which we expect to be reflected in a different mass distribution of these planets when compared to the “normal” planetary population. We use our theoretical survey to produce an upper mass limit for embedded planets: the maximum embedded equilibrium mass (MEEM). This naturally explains the lack of high mass planets between 3 and 64 days orbital period.     

A Systematic Study of the Stability of Symmetric Periodic Orbits in the Planar, Circular, Restricted Three-Body Problem

We investigate symmetric periodic orbits in the framework of the planar, circular, restricted, three-body problem. Having fixed the mass of the primary equal to that of Jupiter, we determine the linear stability of a number of periodic orbits for different values of the eccentricity. A systematic study of internal resonances, with frequency p/q with 2p 9, 1 q 5 and 4/3 p/q 5, offers an overall picture of the stability character of inner orbits. For each resonance we compute the stability of the two possible periodic orbits. A similar analysis is performed for some external periodic orbits.Furthermore, we let the mass of the primary vary and we study the linear stability of the main resonances as a function of the eccentricity and of the mass of the primary. These results lead to interesting conclusions about the stability of exosolar planetary systems. In particular, we study the stability of Earth-like planets in the planetary systems HD168746, GI86, 47UMa,b and HD10697.     

Analytical and numerical results on the stability of a planetary precessional model

A model for planetary precession is investigated using analytical and numerical techniques. A Hamiltonian function governing the model is derived in terms of action-angle Andoyer-Déprit variables under the assumption of equatorial symmetry. As a first approximation a simplified Hamiltonian with zero-eccentricity is considered and stability estimates are derived using KAM theory. A validation of the analytical results is performed computing Poincaré surfaces of section for the circular and elliptical model. We also investigate the role of the eccentricity and its connection with the appearance of resonances. Special attention is devoted to the particular case of the Earth–Moon system. This revised version was published online in July 2006 with corrections to the Cover Date.     

The deep interior of Venus, Mars, and the Earth: A brief review and the need for planetary surface-based measurements

A comparison of the internal structure of Earth-like planets is unavoidable to understand the formation and evolution of the solar system, and the differences between Earth’s, Mars’, and Venus’ atmospheres, surfaces and tectonic behaviors. Recent studies point at the role of core structure and dynamics in the evolution of the atmosphere, mantle and crust. On Earth, the crust thickness and the radius and physical state of the cores are known for almost one century, since the advent of seismological observations, but the lack of long-term surface-based geodetic, electromagnetic and seismological observations on the other planets, results in very large uncertainties on the crust thickness, on the temperature and composition of their mantle, and on the size and physical state of their cores. According to the currently available geodetic data, Mars’ dimensionless mean-moment-of-inertia ratio is equal to 0.3653±0.0008. When combined with geochemical observations and with the inputs of laboratory experiments on planetary materials at high pressure and high temperature, this result constrains a narrow range of density values for Mars’ mantle and favors a light [6200-6765 kg m⁻³] sulfur-rich core, but it still allows for a 1600-1750 km range for the core radius, i.e. an uncertainty at least ten times larger than the precision obtained in 1913 by Gutenberg for the Earth’s core. Mars’ mantle density distribution may be explained by a large range of temperatures and mineralogical compositions, either olivine- or pyroxene-rich. The unknown mean thickness of Mars’ crust makes necessary a number of working assumptions for the interpretation of gravimetric and magnetic data. The situation is worse for Venus, and the most conservative model of its deep interior is a transposition of the Earth’s structure scaled to Venus’ radius and mass. The temperature conditions at the surface of this planet hardly make possible long-term ground-based measurements, but this is indeed feasible at the surface of Mars. Precise measurements of Mars’ crust thickness, core radius and structure, and the proof of the existence or absence of an inner core, would put tight constraints on mantle dynamics and thermal evolution, and on possible scenarios leading to the extinction of Mars’ magnetic field about 4.0 Ga ago. Long-lasting surface-based geodetic, seismological and magnetic observations would provide this information, as well as the distributions as a function of depth of the density, elastic and anelastic parameters, and electrical conductivity. Current studies on the structure of Earth’s deep interior demonstrate that the latter data set, when constrained by laboratory experiments, may be inverted in terms of temperature, chemical, and mineralogical compositions.