Capture probability in the 3:1 mean motion resonance with Jupiter: an application to the Vesta family

We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.     

The two-body interaction potential in the STF tensor formalism: an application to binary asteroids

The symmetric trace free (STF) tensor formalism, developed by Hartmann et al. (Celest Mech Dyn Astron 60:139–159. doi:10.1007/BF00693097, 1994), is a nice tool, not much used in Celestial Mechanics. It is fully equivalent to the usual spherical harmonics but permits more elegant and compact formulations. The coupling between the gravitational fields of extended bodies with this formalism has been used in Mathis and Le Poncin-Lafitte (Astron Astrophys 497:889–910. doi:10.1051/0004-6361/20079054, 2009) for binary stars or planetary systems, but not yet applied to binary asteroids. However, binary asteroids are common in the Solar System and usually their study requires a full two rigid body approach. The formulation of the two-body interaction potential in the STF formalism in the full two rigid body problem is detailed and completed in this article. An application to the binary asteroid (66391) 1999 KW4 is presented with a comparison of our results with other results of the literature for validation.     

Increase of the mean Sun–Earth distance caused by a secular mass accumulation

Based on many planetary observations between the years 1971 and 2003, Krasinsky and Brumberg (Celest. Mech. Dyn. Astron. 90:267–288, 2004) have estimated a rate of increase in the mean Sun-Earth distance of (15±4) m per century. Together with other anomalous observations in the solar system, this increase appears to be unexplained (Lämmerzahl et al. in Astrophys. Space Sci. Lib., vol. 349, pp. 75–101, 2008). We explain these findings by invoking a recently proposed gravitational impact model (Wilhelm et al. in Astrophys. Space Sci. 343:135–144, 2013) that implies a secular mass increase of all massive bodies. This allows us to formulate a quantitative understanding of the effect within the parameter range of the model with a mass accumulation rate of the Sun of (6.4±1.7)×10¹⁰ kg?s^?1.     

Position and velocity sensitivities at the triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate body

In this paper we have examined the stability of triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate spheroid. Here we followed the time limit and computational process of Tuckness (Celest. Mech. Dyn. Mech. 61, 1–19, 1995) on the stability criteria given by McKenzie and Szebehely (Celest. Mech. 23, 223–229, 1981). In this study it was found that in comparison to other studies the value of the critical mass μ _c has been reduced due to oblateness of the bigger primary, i.e. the range of stability of the equilateral triangular libration points reduced with the increase of the oblateness parameter I and hence the order of commensurability was increased.     

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the $J_{2}$ perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.     

Erratum to: Reports of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2006 & 2009

The primary poles for (243) Ida and (134340) Pluto and its satellite (134340) Pluto : I Charon were redefined in the IAU Working Group on Cartographic Coordinates and Rotational Elements (WGCCRE) 2006 report (Seidelmann et al. in Celest Mech Dyn Astr 98:155, 2007), and 2009 report (Archinal et al. in Celest Mech Dyn Astr 109:101, 2011), respectively, to be consistent with the primary poles of similar Solar System bodies. However, the WGCCRE failed to take into account the effect of the redefinition of the poles on the values of the rotation angle W at J2000.0. The revised relationships in Table 3 of Archinal et al. 2011) are $$\begin{array}{llll} W & = & 274^{\circ}.05 +1864^{\circ}.6280070\, d\;{\rm for\; (243)\,Ida} \\ W & = & 302^{\circ} .695 + 56^{\circ} .3625225\, d\;{\rm for\; (134340)\,Pluto,\; and}\\ W & = & 122^{\circ} .695 + 56^{\circ} .3625225\, d\;{\rm for\; (134340)\,Pluto : I \,Charon}\end{array}$$ where d is the time in TDB days from J2000.0 (JD2451545.0).     

On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems

We give a constructive proof of the existence of elliptic lower dimensional tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic Keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytic work in our previous article (Sansottera et al., Celest Mech Dyn Astron 111:337–361, 2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.     

Inclusion of sdBs in evolutionary population synthesis for binary stellar populations and the application: the determination of photo-z and galaxy morphology

Subdwarf B stars (sdBs) can significantly change the ultraviolet spectra of populations at age t～1 Gyr, and have been even included in the evolutionary population synthesis (EPS) models by Han et al. (Mon. Not. R. Astron. Soc. 380:1098, 2007). In this study we present the spectral energy distributions (SEDs) of binary stellar populations (BSPs) by combining the EPS models of Han et al. (Mon. Not. R. Astron. Soc. 380:1098, 2007) and those of the Yunnan group (Zhang et al. in Astron. Astrophys. 415:117, 2004; Mon. Not. R. Astron. Soc. 357:1088, 2005), which have included various binary interactions (except sdBs) in EPS models. This set of SEDs is available upon request from the authors. Using this set of SEDs of BSPs we build the spectra of Burst, E, S0–Sd and Irr types of galaxies by using the package of Bruzual and Charlot (Mon. Not. R. Astron. Soc. 344:1000, 2003). Combined with the photometric data (filters and magnitudes), we obtain the photometric redshifts and morphologies of 1502 galaxies by using the Hyperz code of Bolzonella et al. (Astron. Astrophys. 363:476, 2000). This sample of galaxies is obtained by removing those objects, mismatched with the SDSS/DR7 and GALEX/DR4, from the catalogue of Fukugita et al. (Astron. J. 134:579, 2007). By comparison the results with the SDSS spectroscopic redshifts and the morphological index of Fukugita et al. (Astron. J. 134:579, 2007), we find that the photo-z fluctuate with the SDSS spectroscopic redshifts, while the Sa–Sc galaxies in the catalogue of Fukugita et al. (Astron. J. 134:579, 2007) are classified earlier as Burst-E galaxies.     

Dynamics of Kepler problem with linear drag

We study the dynamics of Kepler problem with linear drag. We prove that motions with nonzero angular momentum have no collisions and travel from infinity to the singularity. In the process, the energy takes all real values and the angular velocity becomes unbounded. We also prove that there are two types of linear motions: capture–collision and ejection–collision. The behaviour of solutions at collisions is the same as in the conservative case. Proofs are obtained using the geometric theory of ordinary differential equations and two regularizations for the singularity of Kepler problem equation. The first, already considered in Diacu (Celest Mech Dyn Astron 75:1–15, 1999), is mainly used for the study of the linear motions. The second, the well known Levi-Civita transformation, allows to complete the study of the asymptotic values of the energy and to prove the existence of collision solutions with arbitrary energy.     

Spectral behavior of AM Her and QQ Vul in high and intermediate states in the UV

We present low resolution UV spectra of two polar systems, AM Her and QQ Vul from the observations taken by the IUE (International Ultraviolet Explorer) of the period between 1978–1996 and 1983–1996 for both systems respectively, to accomplish a large scale study of what happens to the ultraviolet flux of C IV 1550 Å spectral line during different orbital phases. Two spectra for both systems showing the variations in line fluxes and line widths at different orbital phases in high and intermediate states are presented. We concentrated on calculating the line fluxes and line widths of C IV 1550 Å emission line originating in the accretion stream. Our results show that there is spectral variability for the aformentioned physical parameters at different times, similar to that known for the light curve (Heise and Verbunt, Astron. Astrophys. 189:112, 1988; Gansicke et al., Astron. Astrophys. 303:127, 1995; Kafka and Honeycutt, Astron. J. 125:2188K, 2003). We attribute it to the variations of both density and temperature as a result of changing the mass transfer rate (Hutchings et al., Astron. J. 123:2841H, 2002; King and Lasota, Astron. Astrophys. 140L:16K, 1984) which is responsible for this spectral variability. Also we found that the line fluxes of AM Her are greater than the line fluxes of QQ Vul, while the line widths of both systems are approximately the same.     

Secondary resonances and the boundary of effective stability of Trojan motions

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced ‘basic Hamiltonian model’ \(H_\mathrm{b}\) for Trojan dynamics (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez et al. in Celest Mech Dyn Astron 126:519, 2016): we show that the inner border of the secondary resonance of lowermost order, as defined by \(H_\mathrm{b}\), provides a good estimation of the region in phase space for which the orbits remain regular regardless of the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an ‘asymmetric expansion’ of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed.     

Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is \(C^1\) in the velocity. Such an ODE arises as a model of spin–orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on timescales of the order of \(10^6\)–\(10^7\) years. The proposed algorithm is based on the high-order Euler method which was described in Bartuccelli et al. (Celest Mech Dyn Astron 121(3):233–260, 2015), and it requires computer algebra to set up the code for its implementation. The payoff is an overall increase in speed by a factor of about 7.5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.     

Preprocessing the Photospheric Vector Magnetograms for an NLFFF Extrapolation Using a Potential-Field Model and an Optimization Method

Numerical reconstruction/extrapolation of the coronal nonlinear force-free magnetic field (NLFFF) usually takes the photospheric vector magnetogram as input at the bottom boundary. The magnetic field observed at the photosphere, however, contains a force that is in conflict with the fundamental assumption of the force-free model. It also contains measurement noise, which hinders the practical computation. Wiegelmann, Inhester, and Sakurai (Solar Phys. 233, 215, 2006) have proposed to preprocess the raw magnetogram to remove the force and noise to provide better input for NLFFF modeling. In this paper we develop a new code of magnetogram preprocessing that is consistent with our extrapolation method CESE–MHD–NLFFF (Jiang, Feng, and Xiang in Astrophys. J. 755, 62, 2012; Jiang and Feng in Astrophys. J. 749, 135, 2012a). Based on the magnetic-splitting rule that a magnetic field can be split into a potential-field part and a non-potential part, we split the magnetogram and dealt with the two parts separately. The preprocessing of the magnetogram’s potential part is based on a numerical potential-field model, and the non-potential part is preprocessed using the similar optimization method of Wiegelmann, Inhester, and Sakurai (2006). The code was applied to the SDO/HMI data, and results show that the method can remove the force and noise efficiently and improve the extrapolation quality.     

The effects of deformation inertia (kinetic energy) in the orbital and spin evolution of close-in bodies

The purpose of this work is to evaluate the effect of deformation inertia on tide dynamics, particularly within the context of the tide response equations proposed independently by Boué et al. (Celest Mech Dyn Astron 126:31–60, 2016) and Ragazzo and Ruiz (Celest Mech Dyn Astron 128(1):19–59, 2017). The singular limit as the inertia tends to zero is analyzed, and equations for the small inertia regime are proposed. The analysis of Love numbers shows that, independently of the rheology, deformation inertia can be neglected if the tide-forcing frequency is much smaller than the frequency of small oscillations of an ideal body made of a perfect (inviscid) fluid with the same inertial and gravitational properties of the original body. Finally, numerical integration of the full set of equations, which couples tide, spin and orbit, is used to evaluate the effect of inertia on the overall motion. The results are consistent with those obtained from the Love number analysis. The conclusion is that, from the point of view of orbital evolution of celestial bodies, deformation inertia can be safely neglected. (Exceptions may occur when a higher-order harmonic of the tide forcing has a high amplitude.)     

Viscoelastic tides: models for use in Celestial Mechanics

In this note, by using Smale’s \(\alpha \)-theorem on the convergence of Newton’s method, the \(\alpha \)-sets of convergence of some starters of solving the elliptic Kepler’s equation are derived. For each starter we compute the exact \(\alpha \)-set in the eccentricity-main anomaly \((e,M)\in [0,1)\times [0,\pi ]\), showing that these sets are larger than those derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014). Further, new convergence tests based on the Newton–Kantorowitch theorem are given comparing with the derived from Smale’s \(\alpha \)-test.     

Bodily tides near the 1:1 spin-orbit resonance: correction to Goldreich’s dynamical model

Spin-orbit coupling is often described in an approach known as ??the MacDonald torque??, which has long become the textbook standard due to its apparent simplicity. Within this method, a concise expression for the additional tidal potential, derived by MacDonald (Rev Geophys 2:467?C541, 1994), is combined with a convenient assumption that the quality factor Q is frequency-independent (or, equivalently, that the geometric lag angle is constant in time). This makes the treatment unphysical because MacDonald??s derivation of the said formula was, very implicitly, based on keeping the time lag frequency-independent, which is equivalent to setting Q scale as the inverse tidal frequency. This contradiction requires the entire MacDonald treatment of both non-resonant and resonant rotation to be rewritten. The non-resonant case was reconsidered by Efroimsky and Williams (Cel Mech Dyn Astron 104:257?C289, 2009), in application to spin modes distant from the major commensurabilities. In the current paper, we continue this work by introducing the necessary alterations into the MacDonald-torque-based model of falling into a 1-to-1 resonance. (The original version of this model was offered by Goldreich (Astron J 71:1?C7, 1996). Although the MacDonald torque, both in its original formulation and in its corrected version, is incompatible with realistic rheologies of minerals and mantles, it remains a useful toy model, which enables one to obtain, in some situations, qualitatively meaningful results without resorting to the more rigorous (and complicated) theory of Darwin and Kaula. We first address this simplified model in application to an oblate primary body, with tides raised on it by an orbiting zero-inclination secondary. (Here the role of the tidally-perturbed primary can be played by a satellite, the perturbing secondary being its host planet. A planet may as well be the perturbed primary, its host star acting as the tide-raising secondary). We then extend the model to a triaxial primary body experiencing both a tidal and a permanent-figure torque exerted by an orbiting secondary. We consider the effect of the triaxiality on both circulating and librating rotation near the synchronous state. Circulating rotation may evolve toward the libration region or toward a spin faster than synchronous (the so-called pseudosynchronous spin). Which behaviour depends on the orbit eccentricity, the triaxial figure of the primary, and the mass ratio of the secondary and primary bodies. The spin evolution will always stall for the oblate case. For libration with a small amplitude, expressions are derived for the libration frequency, damping rate, and average orientation. Importantly, the stability of pseudosynchronous spin hinges upon the dissipation model. Makarove and Efroimsky (Astrophys J, 2012) have found that a more realistic tidal dissipation model than the corrected MacDonald torque makes pseudosynchronous spin unstable. Besides, for a sufficiently large triaxiality, pseudosynchronism is impossible, no matter what dissipation model is used.     

Analytical invariant manifolds near unstable points and the structure of chaos

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry, Proc Lond Math Soc ser 2, 27:151–170, 1926; Moser, Commun Pure Appl Math 9:673, 1956 and 11:257, 1958; Moser, Giorgilli, Discret Contin Dyn Syst 7:855, 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al., Phys D 29:181, 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice the study of the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain, the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as (i) the order of truncation of the series increases, and (ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.     

On the Use of Meteor Camera Systems in the Detection of Kuiper Belt Objects Through Serendipitous Stellar Occultations

The direct detection of Kuiper Belt Objects (KBOs) by telescopic imaging is not currently practical for objects much less than 100 km in diameter. However, indirect methods such as serendipitous stellar occultations might still be employed to detect these bodies. The method of serendipitous stellar occultations has been previously used with some success in detecting KBOs—Roques et al. (Astron J 132(2):819–822, 2006) detected three Trans-Neptunian objects; Schlichting et al. (Nature 462(7275):895–897, 2009) and Schlichting et al. (Astrophys J 761:150, 2012) each detected a single object in archival Hubble Space Telescope data. However, previous assessments of KBO occultation detection rates have been calculated only for telescopes—we extend this method to video camera systems, and we apply this derivation to the automated meteor camera systems currently in use at the University of Western Ontario. We find that in a typical scenario we can expect one occultation per month. However recent studies such as those of Shankman et al. (Astrophys. J. Lett. 764. doi:10.1088/2041-8205/764/1/L2, 2013) and Gladman et al. (AAS/Division for Planetary Sciences Meeting Abstracts, 2012) which indicate that the population of small KBOs may be smaller than has been assumed in the past may result in a sharp reduction of these rates. Nonetheless, a survey for KBO occultations using existing meteor camera systems may provide valuable information about the number density of KBOs.     

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect.     

Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries

This paper studies the stability of infinitesimal motions about the triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. The perturbation technique developed by Bennet (Icarus 4:177, 1965b) has been used for determination of characteristic exponents. This technique is based on Floquet’s Theory for determination of characteristic exponents in the system with periodic coefficients. The results of the study are analytical and numerical expressions are simulated for the transition curves bounding the region of stability in the μ–e plane, accurate to O(e ²). The unstable region is found to be divided into three parts. The effect of radiation parameter is significant. For small values of e, the results are in favor with the numerical analysis of Danby (Astron. J. 69:166, 1964), Bennet (Icarus 4:177, 1965b), Alfriend and Rand (AIAA J. 6:1024, 1969). The effect of radiation pressure is significant than the oblateness and triaxiality of the primaries.