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Register a new userForbidden zones for circular regular orbits of the Moons in Solar system, R3BP
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Authors
Sergey V. Ershkov
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Previously, we have considered the equations of motion of the three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analyzing such a system of equations, we considered the case of small-body motion of negligible mass around the 2-nd of two giant-bodies (which are rotating around their common centre of masses on Kepler trajectories), the mass of which is assumed to be less than the mass of central body. In the current development, we have derived a key parameter that determines the character of quasi-circular motion of the small 3-rd body relative to the 2-nd body (Planet). Namely, by making several approximations in the equations of motion of the three-body problem, such the system could be reduced to the key governing Riccati-type ordinary differential equations. Under assumptions of R3BP (restricted three-body problem), we additionally note that Riccati-type ODEs above should have the invariant form if the key governing (dimensionless) parameter remains in the range 0.001-0.01. Such an amazing fact let us evaluate the forbidden zones for Moons orbits in the inner Solar system or the zones of the meanings of distances (between Moon-Planet) for which the motion of small body could be predicted to be unstable according to basic features of the solutions of Riccati-type.
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We report integrated orbital fits for the inner regular moons of Neptune based on the most complete astrometric data set to date, with observations from Earth-based telescopes, Voyager 2, and the Hubble Space Telescope covering 1981-2016. We summarize the results in terms of state vectors, mean orbital elements, and orbital uncertainties. The estimated masses of the two innermost moons, Naiad and Thalassa, are $GM_{Naiad}$= 0.0080 $pm$ 0.0043 $km^3 s^{-2}$ and $GM_{Thalassa}$=0.0236 $pm$ 0.0064 $km^3 s^{-2}$, corresponding to densities of 0.80 $pm$ 0.48 $g cm^{-3}$ and 1.23 $pm$ 0.43 $g cm^{-3}$, respectively. Our analysis shows that Naiad and Thalassa are locked in an unusual type of orbital resonance. The resonant argument 73 $dot{lambda}_{Thalassa}$-69 $dot{lambda}_{Naiad}$-4 $dot{Omega}_{Naiad}$ $approx$ 0 librates around 180 deg with an average amplitude of ~66 deg and a period of ~1.9 years for the nominal set of masses. This is the first fourth-order resonance discovered between the moons of the outer planets. More high precision astrometry is needed to better constrain the masses of Naiad and Thalassa, and consequently, the amplitude and the period of libration. We also report on a 13:11 near-resonance of Hippocamp and Proteus, which may lead to a mass estimate of Proteus provided that there are future observations of Hippocamp. Our fit yielded a value for Neptunes oblateness coefficient of $J_2$=3409.1$pm$2.9 $times 10^{-6}$.
The detection of thousands of extrasolar planets by the transit method naturally raises the question of whether potential extrasolar observers could detect the transits of the Solar System planets. We present a comprehensive analysis of the regions in the sky from where transit events of the Solar System planets can be detected. We specify how many different Solar System planets can be observed from any given point in the sky, and find the maximum number to be three. We report the probabilities of a randomly positioned external observer to be able to observe single and multiple Solar System planet transits; specifically, we find a probability of 2.518% to be able to observe at least one transiting planet, 0.229% for at least two transiting planets, and 0.027% for three transiting planets. We identify 68 known exoplanets that have a favourable geometric perspective to allow transit detections in the Solar System and we show how the ongoing K2 mission will extend this list. We use occurrence rates of exoplanets to estimate that there are $3.2pm1.2$ and $6.6^{+1.3}_{-0.8}$ temperate Earth-sized planets orbiting GK and M dwarf stars brighter than $V=13$ and $V=16$ respectively, that are located in the Earths transit zone.
The complex planetary synchronization structure of the solar system, which since Pythagoras of Samos (ca. 570-495 BC) is known as the music of the spheres, is briefly reviewed from the Renaissance up to contemporary research. Copernicus heliocentric model from 1543 suggested that the planets of our solar system form a kind of mutually ordered and quasi-synchronized system. From 1596 to 1619 Kepler formulated preliminary mathematical relations of approximate commensurabilities among the planets, which were later reformulated in the Titius-Bode rule (1766-1772) that successfully predicted the orbital position of Ceres and Uranus. Following the discovery of the ~11 yr sunspot cycle, in 1859 Wolf suggested that the observed solar variability could be approximately synchronized with the orbital movements of Venus, Earth, Jupiter and Saturn. Modern research have further confirmed that: (1) the planetary orbital periods can be approximately deduced from a simple system of resonant frequencies; (2) the solar system oscillates with a specific set of gravitational frequencies, and many of them (e.g. within the range between 3 yr and 100 yr) can be approximately constructed as harmonics of a base period of ~178.38 yr; (3) solar and climate records are also characterized by planetary harmonics from the monthly to the millennia time scales. This short review concludes with an emphasis on the contribution of the authors research on the empirical evidences and physical modeling of both solar and climate variability based on astronomical harmonics. The general conclusion is that the solar system works as a resonator characterized by a specific harmonic planetary structure that synchronizes also the Suns activity and the Earths climate.
Minimal forbidden factors are a useful tool for investigating properties of words and languages. Two factorial languages are distinct if and only if they have different (antifactorial) sets of minimal forbidden factors. There exist algorithms for computing the minimal forbidden factors of a word, as well as of a regular factorial language. Conversely, Crochemore et al. [IPL, 1998] gave an algorithm that, given the trie recognizing a finite antifactorial language $M$, computes a DFA recognizing the language whose set of minimal forbidden factors is $M$. In the same paper, they showed that the obtained DFA is minimal if the input trie recognizes the minimal forbidden factors of a single word. We generalize this result to the case of a circular word. We discuss several combinatorial properties of the minimal forbidden factors of a circular word. As a byproduct, we obtain a formal definition of the factor automaton of a circular word. Finally, we investigate the case of minimal forbidden factors of the circular Fibonacci words.
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