Secular orbit variation due to solar radiation effects: a detailed model for BYORP

A detailed derivation of the effect of solar radiation pressure on the orbit of a body about a primary orbiting the Sun is given. The result is a set of secular equations that can be used for long-term predictions of changes in the orbit. Solar radiation pressure is modeled as a Fourier series in the body’s rotation state, where the coefficients are based on the shape and radiation properties of the body as parameters. In this work, the assumption is made that the body is in a synchronous orbit about the primary and rotates at a constant rate. This model is used to write explicit variational equations of the energy, eccentricity vector, and angular momentum vector for an orbiting body. Given that the effect of the solar radiation pressure and the orbit are periodic functions, they are readily averaged over an orbit. Furthermore, the equations can be averaged again over the orbit of the primary about the Sun to give secular equations for long-term prediction. This methodology is applied to both circular and elliptical orbits, and the full equations for secular changes to the orbit in both cases are presented. These results can be applied to natural systems, such as the binary asteroid system 1999 KW4, to predict their evolution due to the Binary YORP effect, or to artificial Earth orbiting, nadir-pointing satellites to enable more precise models for their orbital evolution.     

Simulation and analysis of the dynamics of binary near-Earth Asteroid (66391) 1999 KW4

Near-Earth Asteroid (66391) 1999 KW4 was the subject of the recently published first extensive radar imaging, shape and mutual orbit modeling, and physical and dynamical characterization of a binary asteroid. In this paper we present in detail our numerical simulation of KW4 behind that work. Our propagations of the system with some variation in estimated parameters cover the set of KW4's possible current dynamical states consistent with the body models and other information obtained directly from the observations. We also apply our implementation of this simulation capability to address some of the dynamical mechanisms by which KW4 may be moved into the more energetically excited of those possible current states, particularly solar gravity interaction. Through comparison of the results with certain features of the observation data, we conclude that the actual KW4 system is not in the most energetically relaxed configuration but must be moderately excited. The system occupies a generalized Cassini state 2 which is different from that considered in most previously published treatments of Cassini states in that it involves co-precession of the primary's spin frame and the mutual orbit rather than co-precession of a satellite's spin frame and that satellite's orbit about the primary. We present a simple analytical theory describing the system's dynamics, which should be applicable to any other binary systems, of which KW4 is representative, in which a massive, roughly oblate primary is spinning rapidly relative to the rate of its mutual orbit with an on-average synchronous, elongated secondary. We examine separately both the effect of the larger binary component's oblateness, and the effect of the smaller component's roughly triaxial ellipsoid shape. The simple analytical formulae obtained agree with full-detail numerical simulation results, and can be used for remote estimation of binary mass properties from observed system motion.     

The Hill stability of inclined small mass binary systems in three-body systems with special application to triple star systems, extrasolar planetary systems and Binary Kuiper Belt systems

The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable.     

Equilibrium points and associated periodic orbits in the gravity of binary asteroid systems: (66391) 1999 KW4 as an example

The motion of a massless particle in the gravity of a binary asteroid system, referred as the restricted full three-body problem (RF3BP), is fundamental, not only for the evolution of the binary system, but also for the design of relevant space missions. In this paper, equilibrium points and associated periodic orbit families in the gravity of a binary system are investigated, with the binary (66391) 1999 KW4 as an example. The polyhedron shape model is used to describe irregular shapes and corresponding gravity fields of the primary and secondary of (66391) 1999 KW4, which is more accurate than the ellipsoid shape model in previous studies and provides a high-fidelity representation of the gravitational environment. Both of the synchronous and non-synchronous states of the binary system are considered. For the synchronous binary system, the equilibrium points and their stability are determined, and periodic orbit families emanating from each equilibrium point are generated by using the shooting (multiple shooting) method and the homotopy method, where the homotopy function connects the circular restricted three-body problem and RF3BP. In the non-synchronous binary system, trajectories of equivalent equilibrium points are calculated, and the associated periodic orbits are obtained by using the homotopy method, where the homotopy function connects the synchronous and non-synchronous systems. Although only the binary (66391) 1999 KW4 is considered, our methods will also be well applicable to other binary systems with polyhedron shape data. Our results on equilibrium points and associated periodic orbits provide general insights into the dynamical environment and orbital behaviors in proximity of small binary asteroids and enable the trajectory design and mission operations in future binary system explorations.     

The Hill stability of a binary or planetary system during encounters with a third inclined body

The dynamical interaction of a binary or planetary system and a third body moving on a parabolic orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in binary star disruption and exchange, in extrasolar planetary system disruption, exchange and capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. This makes exchange or disruption of the component masses more likely as does increasing the eccentricity of the binary.
The stability criteria are applied to determine possible disruption and capture distances for currently known extrasolar planetary systems.     

Rotationally-perturbed orbital elements of close binary systems

The aim of this investigation is to present the secular and periodic perturbations of the six orbital elements of a close binary system due to rotational distortion. In our study we consider very small inclinationst of the orbital plane of the system, whereas the eccentricity of the orbit may assume any value between 0<e<1. The final formulae for the various elements have been expressed by means of the unperturbed true anomaly measured from the ascending node.     

Stability of a planet in the HD 41004 binary system

The Hill stability criterion is applied to analyse the stability of a planet in the binary star system of HD 41004 AB, with the primary and secondary separated by 22 AU, and masses of 0.7 M_⊙ and 0.4 M_⊙, respectively. The primary hosts one planet in an S‐type orbit, and the secondary hosts a brown dwarf (18.64 M_J) on a relatively close orbit, 0.0177 AU, thereby forming another binary pair within this binary system. This star‐brown dwarf pair (HD 41004 B+Bb) is considered a single body during our numerical calculations, while the dynamics of the planet around the primary, HD 41004 Ab, is studied in different phase‐spaces. HD 41004 Ab is a 2.6 M_J planet orbiting at the distance of 1.7 AU with orbital eccentricity 0.39. For the purpose of this study, the system is reduced to a three‐body problem and is solved numerically as the elliptic restricted three‐body problem (ERTBP). The Hill stability function is used as a chaos indicator to configure and analyse the orbital stability of the planet, HD 41004 Ab. The indicator has been effective in measuring the planet's orbital perturbation due to the secondary star during its periastron passage. The calculated Hill stability time series of the planet for the coplanar case shows the stable and quasi‐periodic orbits for at least ten million years. For the reduced ERTBP the stability of the system is also studied for different values of planet's orbital inclination with the binary plane. Also, by recording the planet's ejection time from the system or collision time with a star during the integration period, stability of the system is analysed in a bigger phase‐space of the planet's orbital inclination, ≤ 90°, and its semimajor axis, 1.65–1.75 AU. Based on our analysis it is found that the system can maintain a stable configuration for the planet's orbital inclination as high as 65° relative to the binary plane. The results from the Hill stability criterion and the planet's dynamical lifetime map are found to be consistent with each other. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A timing formula for main-sequence star binary pulsars

In binary radio pulsars with a main-sequence star companion, the spin-induced quadrupole moment of the companion gives rise to a precession of the binary orbit. As a first approximation one can model the secular evolution caused by this classical spin-orbit coupling by linear-in-time changes of the longitude of periastron and the projected semi-major axis of the pulsar orbit. This simple representation of the precession of the orbit neglects two important aspects of the orbital dynamics of a binary pulsar with an oblate companion. First, the quasiperiodic effects along the orbit, owing to the anisotropic 1/ r ³ nature of the quadrupole potential. Secondly, the long-term secular evolution of the binary orbit, which leads to an evolution of the longitude of periastron and the projected semi-major axis, which is non-linear in time.   In this paper a simple timing formula for binary radio pulsars with a main-sequence star companion is presented which models the short-term secular and most of the short-term periodic effects caused by the classical spin-orbit coupling. I also give extensions of the timing formula that account for long-term secular changes in the binary pulsar motion. It is shown that the short-term periodic effects are important for the timing observations of the binary pulsar PSR B1259–63. The long-term secular effects are likely to become important in the next few years of timing observations of the binary pulsar PSR J0045–7319. They could help to restrict or even determine the moments of inertia of the companion star and thus probe its internal structure.   Finally, I reinvestigate the spin-orbit precession of the binary pulsar PSR J0045–7319 since the analysis given in the literature is based on an incorrect expression for the precession of the longitude of periastron. A lower limit of 20° for the inclination of the B star with respect to the orbital plane is derived.     

Formation of the extreme Kuiper-belt binary 2001 QW<Subscript>322</Subscript> through adiabatic switching of orbital elements

Binaries in the Kuiper-belt are unlike all other known binaries in the Solar System. Both their physical and orbital properties are highly unusual and, because these objects are thought to be relics dating back to the earliest days of the Solar System, understanding how they formed may provide valuable insight into the conditions which then prevailed. A number of different mechanisms for the formation of Kuiper-belt binaries (KBBs) have been proposed including; two-body collisions inside the Hill sphere of a larger body; strong dynamical friction; exchange reactions; and chaos assisted capture. So far, no clear consensus has emerged as to which of these mechanisms (if any) can best explain the observed population of KBBs. Indeed, the recent characterization of the mutual orbit of the symmetric (i.e., roughly equal mass) KBB 2001 QW₃₂₂ has only served to complicate the picture because its orbit does not seem readily explicable by any of the available models. The binary 2001 QW₃₂₂ stands out even among the already unusual population of KBBs for the following reasons: its mutual orbit is extremely large (≈10⁵ km or about 30% of the Hill sphere radius), retrograde, it is inclined ≈120^° from the ecliptic and has very low eccentricity, i.e., e ≤ 0.4 (and possibly e ≤ 0.05). Here we propose a hybrid formation mechanism for this object which combines aspects of several of the mechanisms already proposed. Initially two objects are temporarily trapped in a long-living chaotic orbit that lies close to a retrograde periodic orbit in the three-dimensional Hill problem. This is followed by capture through gravitational scattering with a small intruder object. Finally, weak dynamical friction gradually switches the original orbit “adiabatically” into a large, almost circular, retrograde orbit similar to that actually observed.     

Dynamics of close binary systems with mass exchange

The effects of non-isotropic ejection of mass from either component of a binary system on the orbital elements are studied, for the case of a small initial eccentricity of the relative orbit, when all the ejected mass falls on the other component. The problem is transformed to an equivalent two-body problem with isotropic variation of mass, plus a perturbing force which is a function of the intial conditions of ejection of the particles and their final, positions and velocities when they fall on the surface of the other star. The variation of the orbital elements are derived. It is shown that, to first-order terms in the eccentricity, the secular change of the semimajor axis is equal to the one corresponding to the case of zero initial eccentricity. On the contrary, the secular change of the eccentricity is smaller and it depends on the variations of mass ejection due to the finite eccentricity.     

The Hill stability of inclined bound triple star and planetary systems

The dynamical stability of a triple system composed of a binary or planetary system and a bound third body moving on a orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of planetary systems against disruption, component exchange or capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects. These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the third body initially increases the stability of the system then decreases stability as the eccentricity reaches higher values.The Hill stability criterion is applied to extrasolar planetary systems to determine the critical distances at which planets of the same mass as the observed extrasolar planet moving on a circular orbit could remain on a stable orbit. It was found that these distances were sufficiently short suggesting that the presence of further as yet unobserved stable extrasolar planets in observed systems was very likely.     

The Hill stability of a binary or planetary system during encounters with a third inclined body moving on a hyperbolic orbit

The dynamical interaction of a binary or planetary system and a third body moving on a hyperbolic orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in binary star disruption and exchange and planetary system exchange or capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the third body also produces similar effects. These type of changes make exchange or disruption of the component masses more likely as also does increasing the eccentricity of the binary.

The critical distances and Hill stability ranges associated with the possible formation of roughly equal mass trans-Neptunian binaries from three-body interactions are determined for a range of secondary component masses.     

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star–planet–moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.     

Main belt binary asteroidal systems with circular mutual orbits

In 2003, we initiated a long-term Adaptive Optics campaign to study the orbit of various main-belt asteroidal systems. Here we present a consistent solution for the mutual orbits of four binary systems: 22 Kalliope, 45 Eugenia, 107 Camilla and 762 Pulcova. With the exception of 45 Eugenia, we did not detect any additional satellites around these systems although we have the capability of detecting a loosely-bound fragment (located at 1/4×RHill) that is ∼40 times smaller in diameter than the primary. The common characteristic of these mutual orbits is that they are roughly circular. Three of these binary systems belong to a C-“group” taxonomic class. Our estimates of their bulk densities are consistently lower (∼1 g/cm³) than their associated meteorite analogs, suggesting an interior porosity of 30-50% (taking CI-CO meteorites as analogs). 22 Kalliope, a W-type asteroid, has a significantly higher bulk density of ∼3 g/cm³, derived based on IRAS radiometric size measurement. We compare the characteristics of these orbits in the light of tidal-effect evolution.     

Bimodal brightness oscillations in models of young binary systems

We consider a model for the cyclic activity of young binary stars that accrete matter from the remnants of a protostellar cloud. If the orbit of such a binary system is inclined at a small angle to the line of sight, then the streams of matter and the density waves excited in the circumbinary disk can screen the primary component of the binary from the observer. To study these phenomena by the SPH (smoothed particle hydrodynamics) method, we have computed grids of hydrodynamic models for binary systems based on which we have constructed the light curves as a function of the orbital phase. The main emphasis is on investigating the properties of the brightness oscillations. Therefore, the model parameters were varied within the following ranges: the component mass ratio q = M ₂: M ₁ = 0.2–0.5 and the eccentricity = 0–0.7. The parameter that defined the binary viscosity was also varied. We adopted optical grain characteristics typical of circumstellar dust. Our computations have shown that bimodal oscillations are excited in binaries with eccentric orbits, provided that the binary components do not differ too much in mass. In this case, the ratios of the periods and amplitudes of the bimodal oscillations and their shape depend strongly on the inclination of the binary plane and its orientation relative to the observer. Our analysis shows that the computed light curves can be used in interpreting the cyclic activity of UX Ori stars.     

The Hill stability of low mass binaries in hierarchical triple systems

The Hill stability of the low mass binary system in the presence of a massive third body moving on a wider inclined orbit is investigated analytically. It is found that, in the case of the third body being on a nearly circular orbit, the region of Hill stability expands as the binary/third body mass ratio increases and the inclination (i) decreases. This i-dependence decreases very quickly with increasing eccentricity (e ₂) of the third body relative to the binary barycentre. In fact, if e ₂ is not extremely small, the Hill stable region can be approximately expressed in a closed form by setting i = 90°, and it contracts with increasing e ₂ as ${e_2^2}$ for sufficiently low mass binary. Our analytic results are then applied to the observed triple star systems and the Kuiper belt binaries.     

Optimum merger time-scales and binary evolution

A study of galaxy mergers, on the basis of the collisional theory, using galaxy models without halos and considering the evolution of the proginator galaxies only from a time when the gravitational interaction between them is physically significant, indicates that most of the mergers are affected in 2 to 3 orbital periods for progenitors of comparable mass: shorter and longer time-scales being underabundant. These results have a bearing on the evolution of binary galaxies; indicating that once the relative orbit of a binary is circularized, the components will merge during the subsequent orbit or the next one (in a time-scale ~ 10⁸ years). These results are also indicative of the fact that binary evolution is very likely to cause a gradual evolution of the fundamental plane occupied by paired ellipticals from that of isolated ellipticals. After the merger, the remnant is very likely to define a fundamental plane with a slightly different slope.     

The influence of the great inequality on the secular disturbing function of the planetary system

This paper derives the contributionF ₂ ^* by the great inequality to the secular disturbing function of the principal planets. Andoyer's expansion of the planetary disturbing function and von Zeipel's method of eliminating the periodic terms is employed; thereby, the corrected secular disturbing function for the planetary system is derived. An earlier solution suggested by Hill is based on Leverrier's equations for the variation of elements of Jupiter and Saturn and on the semi-empirical adjustment of the coefficients in the secular disturbing function. Nowadays there are several modern methods of eliminating periodic terms from the Hamiltonian and deriving a purely secular disturbing function. Von Zeipel's method is especially suitable. The conclusion is drawn that the canonicity of the equations for the secular variation of the heliocentric elements can be preserved if there be retained, in the secular disturbing function, terms only of the second and fourth order relative to the eccentricity and inclinations.The Krylov-Bogolubov method is suggested for eliminating periodic terms, if it is desired to include the secular perturbations of the fifth and higher order in the heliocentric elements. The additional part of the secular disturbing functionF ₂ ^* derived in this paper can be included in existing theories of the secular effects of principal planets. A better approach would be to preserve the homogeneity of the theory and rederive all the secular perturbations of principal planets using Andoyer's symbolism, including the part produced by the great inequality.     

Orbital evolution of small binary asteroids

About 15% of both near-Earth and main-belt asteroids with diameters below 10 km are now known to be binary. These small asteroid binaries are relatively uniform and typically contain a fast-spinning, flattened primary and a synchronously rotating, elongated secondary that is 20-40% as large (in diameter) as the primary. The principal formation mechanism for these binaries is now thought to be YORP (Yarkovsky-O’Keefe-Radzievskii-Paddack) effect induced spin-up of the primary followed by mass loss and accretion of the secondary from the released material. It has previously been suggested (?uk, M. [2007]. Astrophys. J. 659, L57-L60) that the present population of small binary asteroids is in a steady state between production through YORP and destruction through binary YORP (BYORP), which should increase or decrease secondary’s orbit, depending on the satellite’s shape. However, BYORP-driven evolution has not been directly modeled until now. Here we construct a simple numerical model of the binary’s orbital as well the secondary’s rotational dynamics which includes BYORP and selected terms representing main solar perturbations. We find that many secondaries should be vulnerable to chaotic rotation even for relatively low-eccentricity mutual orbits. We also find that the precession of the mutual orbit for typical small binary asteroids might be dominated by the perturbations from the prolate and librating secondary, rather than the oblate primary. When we evolve the mutual orbit by BYORP we find that the indirect effects on the binary’s eccentricity (through the coupling between the orbit and the secondary’s spin) dominate over direct ones caused by the BYORP acceleration. In particular, outward evolution causes eccentricity to increase and eventually triggers chaotic rotation of the secondary. We conclude that the most likely outcome will be reestablishing of the synchronous lock with a “flipped” secondary which would then evolve back in. For inward evolution we find an initial decrease of eccentricity and secondary’s librations, to be followed by later increase. We think that it is likely that various forms of dissipation we did not model may damp the secondary’s librations close to the primary, allowing for further inward evolution and a possible merger. We conclude that a merger or a tidal disruption of the secondary are the most likely outcomes of the BYORP evolution. Dissociation into heliocentric pairs by BYORP alone should be very difficult, and satellite loss might be restricted to the minority of systems containing more than one satellite at the time.     

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In this paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10⁻³ to 10³. We also present more numerical results for short-term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.