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Published on September 22, 2018

Author: bijay_maniari

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‘moons’ of our planets: ‘moons’ of our planets Bijay Kumar Sharma Ex-Emeritus Fellow NIT, Patna 800003 Fixed Point Solutions of Circular Restricted Three Body Problem. [1] : Fixed Point Solutions of Circular Restricted Three Body Problem . [1] In our Solar System, Sun, Planet and its natural satellite constitutes a Circular Restricted 3-Body Problem CRTBP) which has a fixed point solutions consisting of L1 , L2, L3, L4 and L5. Planet, its respective natural satellite and a test particle also constitutes a CRTBP and it has its corresponding 5 Lagrange Points. Fixed point solutions CRTBP: Fixed point solutions CRTBP Illustration of CRTBP in our Solar System. [2] : Illustration of CRTBP in our Solar System . [2] Primary-Secondary L4 L5 Comments Sun-Earth Asteroid2010TK7 Asteroid2010S016 Trojans of Earth in Earth’s orbit Earth-Moon Kordylewski cloud   Trojan of Moon in Moon’s orbit Sun-Jupiter Dozen asteroids Dozen asteroids Trojans of Jupiter in Jupiter’s orbit. Saturn-Tethys Telesto Calypso Tethys, leading Telesto and lagging Calypso all three are co-orbital, synchronous, orbital period 1.88d Saturn- Dione’s Helene Polydeuces Dione’s, Helene and Polydeuces are co-orbital Sun-Neptune 13Trojans 13 trojans   Sun-Uranus 2011QF99     Sun-Mars 7 Trojans 7 Trojans   Trojans of Neptune [3]: Trojans of Neptune [3] Trojans of Jupiter [4]: Trojans of Jupiter [4] A spike in discovery of irregular moons [5]: A spike in discovery of irregular moons [5] # of irr.Satellites discovered till 2006 [6]: # of irr.Satellites discovered till 2006 [6] Planet #sat, Mp (×10 25 Kg) r min (rad.of sat)(Km) Hill Rad(×10 7 Km) Jupiter 55 190 1 5.1 Saturn 35 57 2 6.9 Uranus 0 8.7 6 7.3 Neptune 7* 10.2 16 11.6 Hill sphere [7]: Hill sphere [7] Gravitational Sphere of Influence (SOI) around a planet in presence of Sun is known as Hill Sphere given by the following formula: A natural satellite is captured within the Hill Sphere surrounding a Planet if the Hill Sphere is spacious enough.   Criteria of ‘moon’ capture by a Planet [8]: Criteria of ‘moon’ capture by a Planet [8] R1=(Planet Radius/Hill Radius) is highly correlated with probability of natural satellite capture. The criteria for capture is : R1 < 0.006 So for a given Planet, semi-major axis of moon ‘ a moon ’ < Hill Radius of the given Planet. Laplace Plane [9]: Laplace Plane [9] For any perturbed orbit of a satellite there exists a Laplace Plane around which the orbital plane of the perturbed orbit precesses . In the inner part, of Planet-satellite system, tidally oblate planet dominates the orbital dynamics and equatorial plane of the planet is the Laplace Plane. In outer part of of the system, Sun’s perturbation dominates the orbital dynamics and the natural satellite orbits in the Ecliptic plane. Laplace Plane Transition [10]: Laplace Plane Transition [10] In between there is Laplace Plane Transition where the oblateness of the planet and Sun’s perturbation balance each other. This is called Laplace Plane Transition ( r L ) orbit. For low obliquity planets the transition is smooth but for high obliquity planet the transition can be complex. Satellites on circular orbit around high obliquity planets migrating through Laplace Plane Transition orbit can acquire substantial eccentricities and inclinations. This has happened in case of our Earth-Moon system. Formalism of Transition [11]: Formalism of Transition [11] Laplace Plane Transition orbit is defined as below: Natural satellites within ‘ r L ’ lie in equatorial plane, they are formed in-situ from the dust accretion disc around the planet and they are called regular satellites.   Formalism of Transition cont’d [12]: Formalism of Transition cont’d [12] satellites beyond ‘ r L ’ lie in ecliptic plane . These satelites are irregular satellites. These are captured bodies. During planetary formation, a lot of small bodies were left over from giant impacts. These were captured from heliocentric orbits into host planets orbits. These are primitive bodies. Different capture mechanisms [13]: Different capture mechanisms [13] captured body from the heliocentric orbit during the early period of Solar System history by Restricted Three Body Problem(RTBP) dynamics but it has a temporary feature. Therefore there must be an auxillary capture mechanism which sets these irregulars in permanent stable orbits. The auxilllary capture mechanisms can be any of the following: capture mechanisms cont’d [14]: capture mechanisms cont’d [14] Through gas-drag capture mechanisms [ Saha and Termaine 1993; Gladman et.al. 2001; Brunini et.al. 2002]. Collisional capture [Colombo & Franklin 1971]; Chaos assisted capture from low energy orbit[ Astakhov et.al. 2003]; Or various binary capture scenarios [ Agnor & Hamilton 2006; Nesvorny et.al.2007; Vokrouhlicky et.al.2008; Gaspar et.al. 2010; Philpott et.al. 2010] Regular Satellites of Jupiter [15]: Regular Satellites of Jupiter [15] Notes on the diagram [16]: Notes on the diagram [16] This figure shows a plan view for the orbits of the regular satellites of Jupiter. They have small circular orbits and low inclinations. These objects probably formed in an early circumjovian disk of gas and dust around Jupiter during Jupiter's formation. Black Dot is Jupiter's location. Purple dotted lines are the orbits of the giant Galilean satellites. Green dashed lines are the orbits of the small inner satellites. Irregular Satellites of Jupiter [17]: Irregular Satellites of Jupiter [17] Notes on the diagram [18]: Notes on the diagram [18] plan view of the orbits of all 31 known outer irregular satellites of Jupiter known before 2002. Irregular satellites have large orbits, inclinations and eccentricities. - Black Dot is Jupiter's location. -Purple dotted line is the orbit of the outer most Galilean satellite Callisto . -Green dotted and dashed line is the inner most irregular prograde satellite Themisto . -Blue dashed lines are the 5 irregular satellites in the prograde group know before 2002. -Red solid lines are the 11 discovered irregular satellites of 2001 in the retrograde group. -Red dashed lines are the 14 previously known irregular satellites in the retrograde group. 12 new moons of Jupiter dis.[18a]: 12 new moons of Jupiter dis.[18a] Explanation of the figure[18c]: Explanation of the figure[18c] Inner blue orbits-2 small sized moons (less than 1km) are orbiting in prograde direction, Outer red orbits- 11 newly discovered moons (all in the range of 1Km dia ) are orbiting in retrograde direction. Outer green orbit- one odd ball called Valetudo is orbiting in prograde direction where general red traffic is in retrograde direction.A collision can occur. Jovian Planets and their respective Hill Radii and Laplace Plane Transition orbits.[19]: Jovian Planets and their respective Hill Radii and Laplace Plane Transition orbits .[19] Planets Earth Jupiter Saturn Uranus Neptune Comments Mass(×10 24 Kg) 5.9723 1898.19 568.34 86.813 102.413   Radius(Km) 6371 69911 58232 25362 24622   †J2 (×10 -6 ) 1082.63 14736 16298 3343.43 3411   ‘a’(×10 6 Km) 149.6 778.57 1433.53 2872.46 4495.06   R H (×10 6 Km) 0.01AU= 1.496280 53.1531 65.4727 70.1294 115.959 respective Hill Radius ‘r L ’(×10 6 Km) 0.0615555 2.3 2.5 1.4 1.8   ‘r L ’(×R Planet ) 10R E (17R E )* 33R J 42.5R S 53R U 73.45R N . Laplace Plane Transition becomes particularly important for planets having high obliquity namely Earth(23.44°),Mars(25.19°),Saturn(26.73°),Uranus(98°),Neptune(28.31°) & Pluto(122.53°). Significance of Laplace Plane Transition[20]: Significance of Laplace Plane Transition[20] All the moons within ‘ r L ’ are under the influence of HOST PLANET hence orbiting in equatorial plane and in synchronous orbits; moons beyond ‘ r L ’ are influenced by solar perturbations and hence are orbiting in Ecliptic plane and are not in synchronous orbits. Regular satellites are Synchronous [21] : Regular satellites are Synchronous [21] Moon-Earth: synchronous; Phobos - Deimos -Mars: synchronous; Galilean Satellites-Jupiter: synchronous; Regular Satellites-Saturn: synchronous; Regular Satellites-Uranus: synchronous; Regular Satellites-Neptune: synchronous; Charon-Plato: triple synchrony since q(mass ratio ) is greater than 0.2 BUT 4 small moons of PLUTO are not synchronous- most surprising. Why is this so? Jupiter and galilean moons [22]: Jupiter and galilean moons [22] Jup . Sat. Ganymede (Galilean) Callisto (Galilean) Io (Galilean) Europa (Galilean) Himalia Metis Mass (Kg) 1.48×10 23 1.08×10 23 8.94×10 22 4.8×10 22 9.56×10 18 9.56×10 16 Rad. (Km) 2631 2400 1815 1569 93 20 (ρ) ( gm /cc) 1.94 1.86 3.55 3.01 2.8 2.8 ‘a’(×10 6 ) Km 1.07 1.883 0.4216 0.6709 11.48 0.128 P spin (d) 7.154553 16.68902 1.769138 3.551181 0.4 0.294779 P orbit (d) 7.154553 16.68902 1.769138 3.551181 250.5662 0.294779 Saturn and regular moons [23]: Saturn and regular moons [23] Saturn Satellites Titan Rhea Iapetus Dione Tethys Enceladus Mimas Mass (Kg) 1.35×10 23 2.49×10 21 1.88 ×10 21 1.05 ×10 21 7.55 ×10 20 8.40 ×10 19 3.8 ×10 19 Rad. (Km) 2575 765 730 560 530 250 196 (ρ) (gm/cc) 1.88 1.33 1.21 1.43 1.21 1.24 1.17 ‘a’(×10 6 ) Km 1.221850 0.527040 3.5613 0.3774 0.29466 0.23802 0.18552 P spin (d) 15.94542 4.517500 79.33018 2.736915 1.887802 1.370218 0.942422 P orbit (d) 15.94542 4.517500 79.33018 2.736915 1.887802 1.370218 0.942422 Uranus and its regular moons [24]: Uranus and its regular moons [24] Uranus Satellites Titania Oberon Umbriel Ariel Miranda Puck Mass (Kg) 3.49×10 21 3.03×10 21 1.27×10 21 1.27×10 21 6.33×10 19 - Rad. (Km) 788.9 761.4 584.7 528.9 235.8 77 (ρ) (gm/cc) 1.7 1.64 1.52 1.56 1.15 - ‘a’(×10 6 ) Km 4.35840 5.826 2.6597 1.9124 1.2978 0.86 P spin (d) 8.705892 13.46324 4.144177 2.520379 1.413479 - P orbit (d) 8.705892 13.46324 4.144177 2.520379 1.413479 0.761832 Neptune and its regular satellites[25]: Neptune and its regular satellites[25] Neptunian Satallites . Triton Proteus Nereid Larrisa Galatea Despina Mass (Kg) 2.14×10 22           Rad. (Km) 1350 200 170 104×89 79 74 (ρ) ( gm /cc) 2.07           ‘a’(×10 6 ) Km 0.3548 0.1176 5.5134 0.0736 0.062 0.0525 P spin (d) -5.87685           P orbit (d) -5.87685 1.122315 360.1362 0.554654 0.428745 0.334655 Pluto and its satellites [26]: Pluto and its satellites [26] Plutonian Satallites . Pluto Charon Styx Nix Kerberos Hydra Mass (Kg) 1.27×10 22 1.9×10 21 - - - - Rad. (Km) 1137 586 5 20 6 20 1 spin per orbit 6.22 spin per orbit 13.6 spin Per orbit 6.04 spin Per orbit 88.9 spin Per orbit ‘a’(×10 6 ) Km 5913.52 from Sun 0.019640 From Pluto 0.042656 From Pluto 0.048694 From Pluto 0.057783 From Pluto 0.064783 From Pluto P spin (d) 6.38725 6.38725 3.239 1.829 5.33 0.4295 P orbit (d) 248.54y Around Sun 6.38725 20.162 24.85 32.168 38.202 Polygonal features in Jupiter [27]: Polygonal features in Jupiter [27] Projected maps of the regions surrounding the north pole (top) and south pole [28]: Projected maps of the regions surrounding the north pole (top) and south pole [28] Unlike Saturn where there is only one cyclonic storm system at each pole, in Jupiter there are 8 polygonal storm system sitting side by side on North pole surrounding a single polar cyclone (the diameter is estimated to be 4000Km to 4600Km) and 5 polygonal storm systems side by side on South Pole surrounding a larger polar cyclone (the diameter is estimated to be 5600Km to 7000Km). Jupiter Magnetosphere [29]: Jupiter Magnetosphere [29] Io (most volcanic in our SS) [30]: Io (most volcanic in our SS) [30] Io - It is the innermost Galilean moon and it is most volcanically active in our whole solar system. It has over 100 active volcanoes that erupt and alter the surface of the moon. The surface looks bright yellow because of sulphur and sulphur compounds. Because of close proximity of Jupiter, Io is subjected to very large tidal forces leading to squeezing and stretching of the interior which in turn causes volcanic activity. Materials thrown up can escape from Io and form plasma torus in Io’s orbit around Jupiter as seen in Figure on slide 29. Saturn’s north polar hexagon features[31]: Saturn’s north polar hexagon features[31] Ring systems of Jovians [32]: Ring systems of Jovians [32] Roche’s Limit [33]: Roche’s Limit [33] The Roche’s Limit roughly divides the domain of rings and satellites though there are numerous exceptions to the rule as shown in Figure on slide 32. Dot-dashed lines are the Roche’s limit. On slide 32. Results of ring particle and nearby satellite interaction [34]: Results of ring particle and nearby satellite interaction [34] Resonantly controlled outer edges of Saturn’s A and B rings; The narrow Encke and Keeler gaps in the outer A ring; Numerous eccentric ringlets at Saturn and Uranus; And the curious arcs embedded in Neptune’s Adam ring Saturn + Phoebe binary pair [35]: Saturn + Phoebe binary pair [35] Phoebe is beyond Laplace Plane Transition orbit. Hence it orbits in ecliptic plane. Phoebe is orbiting in retrograde fashion opposite to the orbital direction of all the other moons. Phoebe is heavily cratered and scarred outpost of Saturn system, about four times farther than Iapetus (nearest major neighbor of Phoebe) . Phoebe continued [36]: Phoebe continued [36] It could be a captured asteroid in a wide eccentric orbit in ecliptic plane. It seems to be one of the original chunks of rocks which precipitated from the solar nebula 4.567Gy ago when the solar system was born. It may be very primitive and one of the KBOs. A lot of projectiles smaller than 100m have hit Phoebe and these projectiles may be from outside or from inside the Saturnian system. The projectiles have chipped off ejecta from Phoebe which have become the retrograde , smaller outermost moons of Saturn. In that sense they are the progeny of Phoebe . Phoebe outlying status [37]: Phoebe outlying status [37]   Comment on Phoebe & Iapetus [38]: Comment on Phoebe & Iapetus [38] Both Phoebe and Iapetus are well within Saturn’s Hill Radius hence they remain captured but they are far enough from oblate Saturn to experience solar perturbation since they are well beyond Laplace Plane Transition orbit of 2.5×10 9 m. Hence they orbit in ecliptic plane and not in equatorial plane of Saturn Photo of Phoebe from Voyager 2 [39]: Photo of Phoebe from Voyager 2 [39] Photo from Cassini at 2000Km [40]: Photo from Cassini at 2000Km [40] Equatorial ridge of Iapetus image from Cassini [41]: Equatorial ridge of Iapetus image from Cassini [41] circum-iapetian disk of dust and ice[42]: circum-iapetian disk of dust and ice[42] Iapetus [43]: Iapetus [43] Three features of Iapetus make it a class apart among the Saturnian moons. These are its present spin period of 79.3days, the present oblate spheroid shape corresponds to the equilibrium figure of a hydrostatic body rotating with a period of 16 hours[ Chandrashekher , S. (1969)] and its equatorial ridge [Castillo- Rogez et.al. 2007]. Iapetus has the largest non-hydrostatic anomaly. Our Moon is the distant second [ Lambeck & Pullan (1980), Garrick- Bethell et.al. (2006)]. Sub-satellite hypothesis [44]: Sub-satellite hypothesis [44] Levison et.al.(2011) put forward the hypothesis of impact generated circum-iapetian disk of dust and ice. This gave birth to sub-satellite beyond Roche’s Limit. This hypothesis explains the three features given on slide 43. Enceladus- Saturn[45] : Enceladus - Saturn[45] Enceladus , a tiny satellite of Saturn , remains an enigma. Its south pole is giving fountains of water Enceladus produce a water plume large enough to drench the whole Saturnian system.     Titan-Saturn [46]: Titan-Saturn [46] Hydro-carbon oceans on Titan [47]: Hydro-carbon oceans on Titan [47] Cassini study has concluded that Titan is a close cousin of Earth but with its own characteristic idiosyncrasies. Titan’s atmosphere and surface behave like Earth- with clouds, rainfall, river valleys and lakes. But instead of water we have hydrocarbon. Titan seasons change in unexpected way very unlike that on Earth. Hubble image of heavily tilted Uranus[48]: Hubble image of heavily tilted Uranus[48] Rings and moons of Uranus [49]: Rings and moons of Uranus [49] Slide 48 shows the rings and moons lying within Laplace Plane Transition orbit r L = 1.4×10 6 Km are constrained by the oblate Uranus into its equatorial plane . The moons Desdemona (62,660 Km) , Juliet (64,360Km), Cressida (61,770Km), Bianca (59.160Km), Portia (66,200 Km), Puck (86,000Km), Belinda (75,260 Km) and Rosalind (69,930 Km) orbiting in the near-vertical equatorial plane of heavily tilted Uranus are seen in (48). Epsilon Ring has a radius of 50,000Km. Irregular Satellites of Uranus [50]: Irregular Satellites of Uranus [50] Sycorx (12,213×10 6 Km) and Caliban (7.169×10 6 Km) both are most outlying moons in Uranus which has Hill Radius(70,129.4×10 6 Km ) and Laplace Plane Transition orbit r L (1.4×10 6 Km). Hence both these outlying moons are in stable orbit around Uranus as they are deep inside the Hill radius. But both these moons are beyond Laplace Plane Transition hence their orbits are strongly dominated by solar perturbation and they are constrained to remain nearer to heliocentric plane namely ecliptic plane. Neptune- Triton [51]: Neptune- Triton [51] Goldreich et.al.(1989) give a more detailed picture of Triton capture. It was a collisional capture with a regular moon of Neptune which resulted in highly eccentric orbit. Eccentricity resulted in tidal dissipation in Triton which resulted in circularization of Triton orbit. Today it is nearly circular in 1 billion years. Triton was molten during tidal evolution and it cannabilized on the regular satellites and perturbed Nereid . This perturbation caused highly eccentric orbit (0.758) and highly inclined orbit (27.6°) of Nereid . The regular satellites within 5R N survived and were constrained to inclined orbits. Irregular Satellites of Neptune [52]: Irregular Satellites of Neptune [52] In the outer part of the Neptunian system there is a population of satellites with various processes of origin. These are irregular satellites and characterized by wide orbits, large inclinations with respect to the equatorial plane of Neptune, large eccentricities and long orbital period. Most probably Neptune didnot capture these large number of small satellites through gas-drag capture mechanisms [ Saha and Termaine 1993; Gladman et.al. 2001; Brunini et.al. 2002 ]. Various capture mechanism in Neptune [53]: Various capture mechanism in Neptune [53] These irregular satellites may have been acquired through the following mechanisms: Collisional capture [Colombo & Franklin 1971]; Chaos assisted capture from low energy orbit[ Astakhov et.al. 2003]; Or various binary capture scenarios [ Agnor & Hamilton 2006; Nesvorny et.al.2007; Vokrouhlicky et.al.2008; Gaspar et.al. 2010; Philpott et.al. 2010] Kozai Resonance: [54]: Kozai Resonance : [54] Kozai resonance has proven to be an important orbit altering mechanism that can bring an outer satellite within the inner part of a Planet’s Hill radius (R H ). Alternatively it can take an irregular satellite outside the Hill sphere and make it free of planetocentric orbit. As a direct consequence of Kozai resonance very few satellites have orbits beyond Neptunian Laplace Plane Transition ( r L = 1.8×10 6 Km) at an inclination with respect to the Eclptic between 50° and 140°. The orbital configuration of the massive inner satellites directly influence the size of the Kozai resonance zone in Hill sphere of a given Planet. Pluto [55]: Pluto [55] New Horizon has shown that Pluto resembles Titan in terms of landscape. As already discussed its four small moon system is very strange and inexplicable. Conclusion [56]: Conclusion [56] The moons and ring systems of the 8 planets and dwarf planet have a mathematical basis for their orbital configurations but much remains to be understood. Here we have given some broad principles. 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