On the Stability Regions of the Trojan Asteroids

The orbits of fictitious bodies around Jupiter’s stable equilibrium points L ₄ and L ₅ were integrated for a fine grid of initial conditions up to 100 million years. We checked the validity of three different dynamical models, namely the spatial, restricted three body problem, a model with Sun, Jupiter and Saturn and also the dynamical model with the Outer Solar System (Jupiter to Neptune). We determined the chaoticity of an orbit with the aid of the Lyapunov Characteristic Exponents (=LCE) and used also a method where the maximum eccentricity of an orbit achieved during the dynamical evolution was examined. The goal of this investigation was to determine the size of the regions of motion around the equilibrium points of Jupiter and to find out the dependance on the inclination of the Trojan’s orbit. Whereas for small inclinations (up to i=20°) the stable regions are almost equally large, for moderate inclinations the size shrinks quite rapidly and disappears completely for i>60^°. Additionally, we found a difference in the dynamics of orbits around L ₄ which – according to the LCE – seem to be more stable than the ones around L ₅.     

Seasonal changes on Jupiter: 1. Factor of activity of the hemispheres

To identify temporal variations of the characteristics of Jupiter’s cloud layer, we take into account the geometric modulation caused by the rotation of the planet and planetary orbital motion. Inclination of the rotation axis to the orbital plane of Jupiter is 3.13°, and the angle between the magnetic axis and the rotation axis is β ≈ 10°. Therefore, over a Jovian year, the jovicentric magnetic declination of the Earth φ _m varies from–13.13° to +13.13°, and the subsolar point on Jupiter’s magnetosphere is shifted by 26.26° per orbital period. In this connection, variations of the Earth’s jovimagnetic latitude on Jupiter will have a prevailing influence in the solar-driven changes of reflective properties of the cloud cover and overcloud haze on Jupiter. Because of the orbit eccentricity (e = 0.048450), the northern hemisphere receives 21% greater solar energy inflow to the atmosphere, because Jupiter is at perihelion near the time of the summer solstice. The results of our studies have shown that the brightness ratio A _j of northern to southern tropical and temperate regions is an evident factor of photometric activity of Jupiter’s atmospheric processes. The analysis of observational data for the period from 1962 to 2015 reveals the existence of cyclic variations of the activity factor A _j of the planetary hemispheres with a period of 11.86 years, which allows us to talk about the seasonal rearrangement of Jupiter’s atmosphere.     

On the character of evolution of orbits in the vicinity of the 1 : 3 Kirkwood gap

The character of orbital evolution for bodies moving near the if 1 : 3 commensurability with Jupiter was studied by model calculations for the time interval of ～500 years. A comparison of oscillations of the orbital elements a, e, q and q′ is made for ensembles of bodies along three starting orbits in the vicinity of the sharp commensurability with Jupiter. These orbits are eccentric ones of low inclinations having perihelia near the Earth's orbit. Examples of a deceleration of the rate of orbital evolution near the sharp commensurability are revealed. The existence of a group of asteroids connected with the Kirkwood gap, i.e., being in a resonant motion with Jupiter, is suggested. A connection of asteroids 887 Alinda and 1915 Quetzalcoatl with this gap is confirmed.     

The 12C/13C ratio in jupiter from the Voyager infrared investigation

The ¹²C/¹³C ratio in Jupiter has been derived from the analysis of the ν₄ band of CH₄ in the spectra recorded by the Voyager 1 IRIS experiment. It is found to be 160_?55⁺⁴⁰, i.e., 1.8_?0.6^+0.4 times the terrestrial value. Instrumental noise as well as systematic sources of error were taken into account for the estimate of the uncertainty. No plausible theory predicts such a difference between the values of the ¹²C/¹³C ratio in the inner solar system and in Jupiter. However, values of this ratio in the solar neighborhood 4.5 by ago inferred—through the use of models of chemical evolution of the Galaxy —from recent interstellar medium measurements are compatible with the present determination in Jupiter. The Jovian value, rather than the terrestrial one, could then be representative of the ratio in the primitive solar nebula.     

A study of the 3ν3CH4 region in a high-resolution spectrum of Jupiter recorded by Fourier transform spectroscopy

A spectrum of Jupiter between 6000 and 12 000 cm^? at high resolution (0.05 cm^?) was recorded with a Michelson interferometer at Palomar Mountain in October 1974. An analysis of the R branch of the 3ν₃CH₄ band with the reflecting-layer model, taking into account the H₂ absorption which occurs in the same spectral range, leads to a Lorentzian half-width of 0.09 ± 0.02 cm^?1, a rotational temperature of 175 ± 10° K, and a CH₄ abundance of order 52m atm. Five lines of the ¹³CH₄3ν₃ band have been identified; a comparison with new laboratory spectra indicates that the ¹³CH₄/¹²CH₄ ratio in the Jupiter atmosphere is close to the terrestrial ratio.     

For infrared spectrophotometry of Jupiter and Saturn

Infrared spectral observations of Mars, Jupiter, and Saturn were made from 100 to 470 cm^?1 using NASA's G. P. Kuiper Airborne Observatory. Taking Mars as a calibration source, we determined brightness temperatures of Jupiter and Saturn with approximately 5 cm^?1 resolution. The data are used to determine the internal luminosities of the giant planets, for which more than 75% of the thermally emitted power is estimated to be in the measured bandpass: for Jupiter L_J = (8.0 ± 2.0) × 10^?10L_⊙ and for Saturn L_S = (3.6 ± 0.9) × 10^?10. The ratio R of thermally emitted power to solar power absorbed was estimated to be R_J = 1.6 ± 0.2, and R_S = 2.7 ± 0.8 from the observations when both planets were near perihelion. The Jupiter spectrum clearly shows the presence of the rotational ammonia transitions which strongly influence the opacity at frequencies ?250 cm^?1. Comparison of the data with spectra predicted from current models of Jupiter and Saturn permits inferences regarding the structure of the planetary atmospheres below the temperature inversion. In particular, an opacity source in addition to gaseous hydrogen and ammonia, such as ammonia ice crystals as suggested by Orton, may be necessary to explain the observed Jupiter spectrum in the vicinity of 250 cm^?1.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

The effect of dense cores on the structure and evolution of Jupiter and Saturn

We have calculated evolutionary and static models of Jupiter and Saturn with homogeneous solar composition mantles and dense cores of material consisting of solar abundances of SiO₂, MgO, Fe, and Ni. Evolutionary sequences for Jupiter were calculated with cores of mass 2, 4, 6, and 8% of the Jovian mass. Evolutionary sequences for Saturn were calculated with cores of mass 16, 18, 20, and 22% of total mass. Two envelope mixtures, representative of the solar abundances were used: X (mass fraction of hydrogen) = 0.74, Y (mass fraction of helium) = 0.24 and X = 0.77 and Y = 0.21. For Jupiter, the observations of the temperature at 1 bar pressure (T_1bar), radius and internal luminosity were best fit by evolutionary models with a core mass of ～6.5% and chemical composition of X = 0.77, Y = 0.21. The calculated cooling time for Jupiter is approximately 4.9 × 10⁹ years, which is consistent, within our error bars, with the known age of the solar system. For Saturn, the observations of the radius, internal luminosity and T_1BAR can be best fit by evolutionary models with a core mass of ～21% and chemical composition of X = 0.77, Y = 0.21. The cooling time calculated for Saturn is approximately 2.6 × 10⁹ years, almost a factor 2 less than the present age of the solar system. Static models of Jupiter and Saturn were calculated for the above chemical compositions in order to investigate the sensitivity of the calculated gravitational moments, J₂ and J₄, to the mass of the dense core, T_1BAR and hydrogen/helium ratio. We find for Jupiter that a model having a core mass of approximately 7% gives values of J₂, J₄, and T_1BAR that are within observational limits, for the mixture X = 0.77, Y = 0.21. The static Jupiter models are completely consistent with the evolutionary results. For Saturn, the quantities J₂, J₄, and J₆ determined from the static models with the most probable T_1BAR of 140°K, using modeling procedures which result in consistent models for Jupiter, are considerably below the observed values.     

Motion in the generalized restricted three-body problem

This paper investigates the motion of an infinitesimal body in the generalized restricted three-body problem. It is generalized in the sense that both primaries are radiating, oblate bodies, together with the effect of gravitational potential from a belt. It derives equations of the motion, locates positions of the equilibrium points and examines their linear stability. It has been found that, in addition to the usual five equilibrium points, there appear two new collinear points L _n1, L _n2 due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points L ₁, L ₃ come nearer to the primaries; while L ₂, L ₄, L ₅, L _n1 move towards the less massive primary and L _n2 moves away from it. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ _c and unstable for $\mu_{c} \le\mu\le\frac{1}{2}$ , where μ _c is the critical mass ratio influenced by the oblateness and radiation of the primaries and potential from the belt, all of which have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near the oblate, radiating binary stars systems surrounded by a belt.     

The atmospheric composition and structure of Jupiter and Saturn from ISO observations: a preliminary review

Infrared spectra of Jupiter and Saturn have been recorded with the two spectrometers of the Infrared Space Observatory (ISO) in 1995–1998, in the 2.3–180 μm range. Both the grating modes (R=150–2000) and the Fabry-Pérot modes (R=8000–30,000) of the two instruments were used. The main results of these observations are (1) the detection of water vapour in the deep troposphere of Saturn; (2) the detection of new hydrocarbons (CH₃C₂H, C₄H₂, C₆H₆, CH₃) in Saturn’s stratosphere; (3) the detection of water vapour and carbon dioxide in the stratospheres of Jupiter and Saturn; (4) a new determination of the D/H ratio from the detection of HD rotational lines. The origin of the external oxygen source on Jupiter and Saturn (also found in the other giant planets and Titan in comparable amounts) may be either interplanetary (micrometeoritic flux) or local (rings and/or satellites). The D/H determination in Jupiter, comparable to Saturn’s result, is in agreement with the recent measurement by the Galileo probe (Mahaffy, P.R., Donahue, T.M., Atreya, S.K., Owen, T.C., Niemann, H.B., 1998. Galileo probe measurements of D/H and ³He/⁴He in Jupiters atmosphere. Space Science Rev. 84 251–263); the D/H values on Uranus and Neptune are significantly higher, as expected from current models of planetary formation.     

Broadband Submillimeter Spectroscopy of HCN, NH3, and PH3in the Troposphere of Jupiter

We report measurements of the Jupiter brightness spectrum in the 850-μm and 1100-μm atmospheric windows with a spectral resolution of 125 MHz, obtained with a Fourier transform spectrometer on the James Clerk Maxwell Telescope. Three results were obtained. First, the predicted absorption features due to the rotational lines of HCN at 266 and 354 GHz were not detected within our error limits of less than 1%. We establish new upper limits for the HCN abundance in the jovian troposphere for five assumed abundance distributions and for two assumed NH₃abundances. The upper limits are 1.7 to 13 times smaller than the abundance value obtained in the only reported detection of HCN in Jupiter prior to the impact of Shoemaker–Levy 9. Second, the continuum brightness temperature spectrum at 850 μm was determined and is in agreement with previous measurements, but has large error bars due to uncertainties in the photometric calibration. We estimate the ammonia abundance in the 1–2 bar region to be 1.7 times solar, but this result is tentative since scattering by NH₃cloud particles and absorption by gaseous H₂S were neglected in our atmospheric model. Finally, the first rotational line of PH₃at 267 GHz was not detected, a result which we demonstrate is consistent with the statistical noise level in these measurements, with current values of the spectroscopic parameters, and with phosphine measurements at other wavelengths.     

Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5: 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravitational forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simultaneous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6″/yr and the period T _J = T _S ≈ 50687 yr. These results are close to those obtained in the general theory (25.93″/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β₁ ≈ 0.8403504° (Saturn) and β₂ ≈ 0.3409296° (Jupiter) around the point which is located at an angular distance of 1.647607° from the ecliptic pole.     

On the C/H and D/H ratios in the atmospheres of Jupiter and Saturn

Using a method defined in a previous paper [M. Combes and T. Encrenaz, Icarus39 1–27 (1979)], we reestimated the C/H ratio in the atmospheres of Jupiter and Saturn by the measurements of the weak visible CH₄ bands, the CH₄ 3ν₃ band, and the (3-0) and (4-0) quadrupole bands of H₂. In the case of Jupiter we conclude that the C/H ratio is enriched by a factor ranging from 1.7 to 3.6 relative to the solar value. In the case of Saturn, our derived C/H value ranges from 1.2 to 3.2 times the solar value. The Jovian D/H ratio derived from this study is 1.2 × 10^?5 < D/H < 3.1 × 10^?5. The value derived for the D/H ratio on Saturn is not precise enough to be conclusive.     

A survey of orbits of co-orbitals of Mars

Many asteroids with a semimajor axis close to that of Mars have been discovered in the last several years. Potentially some of these could be in 1:1 resonance with Mars, much as are the classic Trojan asteroids with Jupiter, and its lesser-known horseshoe companions with Earth. In the 1990s, two Trojan companions of Mars, 5261 Eureka and 1998 VF₃₁, were discovered, librating about the L₅ Lagrange point, 60° behind Mars in its orbit. Although several other potential Mars Trojans have been identified, our orbital calculations show only one other known asteroid, 1999 UJ₇, to be a Trojan, associated with the L₄ Lagrange point, 60° ahead of Mars in its orbit. We further find that asteroid 36017 (1999 ND₄₃) is a horseshoe librator, alternating with periods of Trojan motion. This asteroid makes repeated close approaches to Earth and has a chaotic orbit whose behavior can be confidently predicted for less than 3000 years. We identify two objects, 2001 HW₁₅ and 2000 TG₂, within the resonant region capable of undergoing what we designate “circulation transition”, in which objects can pass between circulation outside the orbit of Mars and circulation inside it, or vice versa. The eccentricity of the orbit of Mars appears to play an important role in circulation transition and in horseshoe motion. Based on the orbits and on spectroscopic data, the Trojan asteroids of Mars may be primordial bodies, while some co-orbital bodies may be in a temporary state of motion.     

Particle motion in the Schwarzschild-Quintessence space-time

The influence of free static spherically symmetric quintessence on particle motion in the Schwarzschild-quintessence space-time has been studied by numerical calculation. In the Schwarzschild space-time, the particle motion can be determined by an effective potential. However, this potential is dependent on the quintessence’s state parameter w _q . We find that when the quintessence’s state parameter w _q is in the range of $[-\frac{1}{3},0]$ , the massive particle’s motion is just like that in the Schwarzschild space-time. And when $-1\leqslant w_{q}<-\frac{1}{3}$ , a maximum unstable circular orbit exists for every L, and no matter how small L is, the scattering state exists, which leads to the accelerating expansion of our universe. The exists of the maximum orbit can even explain why galaxies is in a ball.     

Interplanetary dust particles near Jupiter

We calculate the expected counting rate of a flat micrometeoroid detector of finite sensitivity passing in hyperbolic orbit near a planet. We assume that the distribution of particle sizes, s, can be expressed as a power law spectrum of index p, i.e. dN(s) = Cs^?pds, and also that the particles encounter the sphere of influence of the planet with a certain speed v_∞. The results of the calculations are then compared with the results returned by Pioneer 10 in its flyby of Jupiter. The observed increase in impact rate near Jupiter can be completely explained in terms of gravitational “focusing” of particles which are in heliocentric orbits; i.e., they are not in orbit about Jupiter. The absolute concentration of particles near the orbit of Jupiter is of the same order as at 1 AU: the exact ratio being a function of particle speed and spectral index. Data from one flyby are insufficient to determine a unique value for both the spectral index, p, and the particle velocity, v_∞, but limits can be set. For reasonable encounter speeds (corresponding to eccentricities and inclinations of dust particles experienced near the Earth), the particles near Jupiter are characterized by a spectrum of index p ～ 3. The spectral index which best fits the data increases with increasing encounter speeds.     

Long term evolution of quasi-circular Trojan orbits

Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of ${\sqrt{\mu}}$ . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L₄ and L₅) and an unstable point at 180° (L₃). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.     

A perturbative treatment of motion near the 3/1 commensurability

A semianalytic perturbation theory for motion near the 3/1 commensurability in the planar elliptic restricted three-body problem is presented. The predictions of the theory are in good agreement with the features found on numerically generated surfaces of section; a global understanding of the phase space is achieved. The unusual features of the motion discovered by J. Wisdom (1982, Astron. J.87, 577–593; 1983a, Icarus56, 51–74) are explained. The principal cause of the large chaotic zone near the 3/1 commensurability is identified, and a new criterion for the existence of large-scale chaotic behavior is presented.     

Orbital resonance in a dissipative medium

Orbital resonances tend to force bodies into noncircular orbits. If a body is also under the influence of an eccentricity-reducing medium, it will experience a secular change in semimajor axis which may be positive or negative depending on whether its orbit is exterior or interior to that of the perturbing body. Thus a dissipative medium can promote either a loss or a gain in orbital energy. This process may explain the resonant structure of the asteroid belt and of Saturn's rings. For reasonable early solar system parameters, it would clear a gap near the 2:1 resonance with Jupiter on a time scale of a few thousand years; the gap width would be comparable to the Kirkwood gap presently at the location in the asteroid belt. Similarly, a gap comparable in width to Cassini's division would be cleared in Saturn's rings at the 2:1 resonance with Mimas in ～10⁶ yr. Most of the material from the gap would be deposited at the outer edge of ring B. The process would also affect the radial distribution of preplanetary material. Moreover, it provides an explanation for the large amplitude of the Titan-Hyperion libration. Consideration of the effects of dissipation on orbits near the stable L₄ and L₅ points of the restricted three-body problem indicates that energy loss causes particles to move away from these points. This results explains the large amplitude of Trojan asteroids about these points and the possible capture of Trojan into orbit about Jupiter.