Anna Maria Nobili

Dynamics of planet-crossing asteroids: Classes of orbital behavior: Project SPACEGUARD

Abstract We present the first results of the SPACEGUARD numerical integration of 410 planet-crossing asteroids spanning 200,000 years. All the planets except Mercury and Pluto are taken into account. The SPACEGUARD data base includes the orbital elements of all the asteroids and planets (both sampled and filtered) and the positions at each close approach. We report a preliminary analysis of the Earth-crossing and almost-crossing asteroids, 89 in our sample. By means of a graphic representation of the orbital evolution these can be divided into six well-defined classes of dynamical behavior, named after the best-known, most representative object in each class: Geographos, Toro, Alinda, Kozai, Oljato, and Eros. This classification replaces the previous Aten-Apollo-Amor one, based upon the osculating elements and, therefore, not indicative of long-term behavior. On the contrary the six SPACEGUARD classes are much more stable: some transitions are observed and they can be used to describe the evolution of the corresponding populations over longer time spans. The asteroids in mean motion resonances with the Earth (Toro class) can be protected against close approaches for a few tens of thousands years, then they return to the nonresonant Earth-approaching state (Geographos class). Mean motion resonances with Jupiter (Alinda class) can change the eccentricity to Earth-crossing values and are, therefore, established as a direct source of meteorites. However, they can also change the eccentricity to such extreme values that transitions to and from large-scale chaotic orbits (Oljato class) can occur. This shows that objects found in a Kirkwood gap could have originated in the outer Solar System. The almost Earth-crossing asteroids can either have perihelion permanently above 1 AU (Eros class) or be protected from node crossing by secular resonances and e-ω coupling (Kozai class). Among the objects which have been observed and classified as planet-crossing asteroids we find also typical cometary orbits.

Dynamics of Pluto

Abstract We report the results of our investigation of the orbit of Pluto as obtained from the LONGSTOP 1B numerical integration of the outer planets spanning 100 Myr. We have analyzed all the critical arguments associated with the 3 : 2 resonance in mean motion with Neptune up to degree 2 in the eccentricities and inclinations. The longitude libration with a period of about 19,900 years is confirmed; the librating argument has a very large (84°) and remarkably constant maximum excursion. The 3.78-Myr period libration of the argument of pericenter of Pluto is also confirmed. A third resonance belonging to an entirely new class of super resonances is discovered; it results from a 1:1 locking between a circulating and a librating secular argument and has a libration period of 34.5 Myr. The three resonance locking do not undergo any significant changes over the LONGSTOP 1B 100-Myr time span or over the 845-Myr span of the Digital Orrery numerical integration. Thus, the Orrery finding of a positive Lyapounov exponent for Pluto still calls for a plausible dynamical explanation. The distance between two initially nearby Plutos grows exponentially up to a saturation distance which simply reflects the maximum excursion of the longitude libration. Since chaos is known to be originated by small divisors, we have looked for possible resonances between the frequencies associated with the critical arguments. The only possible small divisor which is not of exceedingly high order is associated with a 3:1 super resonance ; in the LONGSTOP 1B solution the corresponding argument circulates with a period of ≃246Myr. Small divisors are very sensitive to the values of the planetary masses, and it turns out that the Orrery value of this small divisor is essentially zero (mostly because of the different value they used for the mass of Neptune). We conjecture that the positive Lyapounov exponent might be due to the Orrery solution being locked in this further resonance. If this is so, then the value of the Lyapounov exponent could be very sensitive to the initial conditions and masses. Moreover, the macroscopic stability of the orbit of Pluto would be explained because high-order resonances are associated with small chaotic regions. A more challenging task is to explain how Pluto was locked in such a complicated system of three resonances.

Hyperion: Collisional disruption of a resonant satellite☆

Abstract Hyperion is an irregularly shaped object of about 285 km in mean diameter, which appears as the likely remmant of a catastrophic collisional evolution. Since the peculiar orbit of this satellite (in 4 3 resonance locking with Titan) provides an effective mechanism to prevent any reaccretion of secondary fragments originated in a breakup event, the present Hyperion is probably the “core” of a disrupted precursor. This contrasts with the other, regularly shaped small satellites of Saturn, which, according to B.A. Smith et al. [Science 215, 504–537 (1982)], were disrupted several times but could reaccrete from narrow rings of collisional fragments. The numerical experiments performed to explore the region of the phase space surrounding the present orbit show that most fragments ejected with a relative velocity ⪸0.1 km / sec rapidly attain chaotic-type orbits, having repeated close encounters with Titan. Ejection velocities of this order of magnitude are indeed expected for a collision at a velocity of ∼ 10 km/sec with a projectile-to-target mass ratio of the order of 10−3; similar effects could be produced by less energetic but nearly grazing collisions. Such events are not likely to displace the largest remnant (i.e., the present Hyperion) outside the stable region of the phase space associated with the resonance, but could be responsible for the large amplitude of the observed orbital libration.

Tidal evolution and the Pluto-Charon system

We analyze the system formed by Pluto and its satellite Charon from the point of view of the theory of tidal evolution. The singular feature of the system, i.e. the configuration of complete synchronism which has been suggested by the available data, is found to represent the stable end-product of the evolution. The time needed for the synchronization is shown to be less than the age of the solar system, provided that Plutos tidal dissipation function is smaller than 104–105. Moreover, the initial orbital radius of the system could not be largerthan two or three times the present radius, so that Charon has been always a close satellite.Finally, we discuss Lyttletons hypothesis that Pluto is an escaped satellite of Neptune, suggesting that a possible mechanism of Plutos ejection could be connected with a retrograde capture of Triton by Neptune or with the subsequent tidal evolution of Tritons orbit.

Some remarks on the capture of Triton and the origin of Pluto

Abstract Harrington and Van Flandern (1979, Icarus 39, 131–136) suggests that the irregular features of the Neptunian satellite system and Plutos escape were caused by an encounter with a massive external body. They rule out the alternative mechanism based on the capture of Triton (which seems more plausible because it does not appeal to any unobserved object) on the basis of an incorrect deduction from McCords (1966, Astron. J. 71, 585–590) analysis on the tidal decay of Tritons orbit. As a matter of fact, many recent results show that satellite captures are possible, and in the case of Triton several arguments support this interpretation.

'Galileo Galilei' (GG): space test of the weak equivalence principle to 10^(−17) and laboratory demonstrations

The small satellite ‘Galileo Galilei’ (GG) will test the universality of free fall and hence the weak equivalence principle which is the founding pillar of general relativity to 1 part in 10 17 . It will use proof masses whose atoms differ substantially from one another in their mass energy content, so as to maximize the chance of violation. GG will improve by four orders of magnitude the current best ‘E¨ ot-Wash’ tests based on slowly rotating torsion balances, which have been able to reach their thermal noise level. In GG, the expected violation signal is a relative displacement between the proof masses of � 0.6 pm caused by a differential acceleration aGG � 8 × 10 −17 ms −2 pointing to the center of mass of the Earth as the satellite orbits around it at νGG � 1.7 × 10 −4 Hz. GG will fly an innovative acceleration sensor based on rapidly rotating macroscopic test masses weakly coupled in 2D which up-converts the signal to νspin � 1H z, a value well above the frequency of natural oscillations of the masses relative to each other νd = 1/Td � 1/(540 s). The sensor is unique in that it ensures high rotation frequency, low thermal noise and no attenuation of the signal strength (Pegna et al 2011 Phys. Rev. Lett. 107 200801). A readout based on a very

On topological stability in the general three-body problem

The confining curves in the general three-body problem are studied; the role of the integralc2h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio ε=m3/m2 both the positions of the collinear equilibrium points and thec2h integral are expanded in power series of ε. In this way the relationship is found between the confining curves resulting from thec2h integral in the general problem, and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of ε the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual, system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small ε, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be ‘unstable’, i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that given for the most important three-body subsystems of the Solar System essentially the same information on ‘stability’ as the full computation of thec2h integral and of the bifurcation values.

On the stability of hierarchical four-body systems

AbstractGeneralized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c2h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL2 collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN≥4 bodies, and the main confinement condition, the Easton inequality, can also be extended but it no longer provides separate regions of trapped motion, whatever is the value ofz for the wholeN-body system,N≥4. However, thez criterion of hierarchical stability applies to every 3-body subsystem, whosez ‘integral’ will of course vary in time because of the perturbations from the other bodies. In theN=4 case we decompose the system into two 3-body subsystems whosec2h ‘integrals’,z23 andz34, att=0 are assumed to be smaller than the corresponding critical values

Instability of the 2+2 body problem

\tilde z_{23}