Odessa Astronoical Publications, vol. 8/ (05) 69 ON THE HEURISTIC RULE FOR PLANETARY DISTANCE DISTRIBUTION N. D. Svyazhyn I.I.Mechnikov Odessa National University Odessa 6508, Ukraine, swjashin@onu.edu.ua ABSTRACT. This paper presents a new heuristic rule for the planetary distance distribution in the solar syste siilar to the Titius-Bode rule of planetary orbit spacing. Application of this universal rule siultaneously for planets and planetary oons has been considered. Natural satellites orbiting around a central body are divided into s of six satellites in each.. Introduction There is a vast literature on the search of regularities of planetary and oon orbit spacing according to the Titius- Bode type relation [ 8]. The Titius-Bode Law of Planetary Distances: Its History and Theory by Michael M. Nieto fro the Niels Bohr Institute at the University of Copenhagen was issued in 97. Apparently, the Titius- Bode relation expresses, to soe extent, Newtonian echanics in epirical for: each planet in the solar syste is about.7 ties further fro the Sun than the next innerost planet. It was also shown that such regularities are realised in exoplanetary systes [, 7]. The geoetric series for distances follows fro Newton s law; however, to perfor sufficient siulation and deepen understanding of this phenoenon, it is necessary to rely on the ethods of celestial echanics and apply odern coputer technologies. This study presents a new heuristic rule for the spacing of systes of different bodies in the solar syste.. Rule definition Natural satellites orbiting around a central body are divided into s of six oons in each: h n, n.6 () where the ; n the ordinal of a oon starting with the central body; h n the average distance between the central body and oon which equals to the radius of a sphere which has the sae area as the planar figure restricted by the oon s orbit. If a, b the ellipse sei axes, then the sought radius equals to ab. The distances in the are approxiated with the following forulae (see Table ). (Here the non diensional paraeter; H the average orbital radius of the 6 th oon in the, which is called the upper boundary of the and h the lower boundary of a. The distances h n within s of oons are related as follows: k h k n h n, k 0,,,3... () where +k the of a ). Having the relative values entered, the previous table can be presented as follows (see Table ). As is evident, here a n h n and h 6 h 6 (3) h 3 Table Moon 3 5 6 Distance notation h h h 3 h h 5 h 6 Distance forula H H H H H H Table Moon a a a 3 a a 5 a 6 Distance forula
70 Odessa Astronoical Publications, vol. 8/ (05) Besides, it is supposed that the sequencing axio is realised for planetary distances with any allowed values of the paraeter : 0 a n h n a, n,5 n h 6 fro which the left restriction for the paraeter is obtained: () (5) Fro the sequencing axio, which states that the upper boundary of the is less than the lower boundary of the next, h, n,,3,...6 6 h n the right restriction is obtained: (6) (7) Let us suppose that there are two Phaetons rather than one hidden in the asteroid belt at the distances of.6 and.9 AU fro the Sun, and that asteroid Chiron (a = 3.65 AU, e = 0.38) is a inor planet (or its reainder). According to the above forulated rule we receive the following (see Table 3). Asteroid 538P L with the average orbital radius of.6763 AU, asteroid 99DT with the average radius of.903035 AU and asteroid 999W0 with the average radius.93997 AU (or the ebers of the Flora and Eos failies) were selected as the fragents of the 5 th and 6 th planets within the first. Thus, the values.9,,, 3; β 3 rather accurately approxiate the relative distances h k, k.6,.3, obtained on the basis of actual h 6 data. The ajority of the Kuiper belt asteroids are in the region extending between the orbits of the last planet of the second and the second planet of the third. The 3 rd, th, 5 th and 6 th planets of the third aong trans Neptunian objects do not belong to failies. 3. Description of oon systes To ake it ore illustrative, it is ore convenient to exaine the Neptunian oon syste first (see Table ). As is seen fro Table, the first four oons ake up a faily of the first object within the first. Positions for the th and 5 th objects within the first are epty. The second is copletely epty. The existence of positions in the second is deterined by the values β, which should eet soe additional requireents (see Forula 8). Besides, a definite rule, such as density axio, can be set: in accordance with this axio paraeters α and β should take on the least values given that conditions () and (6) are fulfilled. In other words, when distributing oons within the first and third s, the existence of the second which fulfils condition (6) follows fro condition (). Table 3 3 h 3 h PLANET β = 3. AVERAGE RADIUS RELATION WITHIN A GROUP THEORETICAL RELATION Mercury.38903.30.30 Venus.733359.6.5 3 Earth.999930.30.30 Mars.50375.57.56 5 The Flora faily (.66567).77.77 6 The Eos faily (.9). Jupiter 5.08709.3.3 3.6088 Saturn 9.53007.5.5 3.759 3 Chiron 3.0883.3.3 3.0975 Uranus 9.8058.9.5.6608 5 Neptune 30.06809.77.77 3.588 6 Pluto 38.855936.. 3.35555 (976) 007 UK 68.3088 68.69.39.3 3. (890) 999 RD, 05.56..8393 (858) 00FP 57.7.39.5.08 3 (809) 009 CR 73.0357 69.6.3.3.85 00 VN 5.656 3.365.93.5.69 5 (90377) Sedna 393.8966 37.63.76.77.6 6 006 SQ 509.076 93...69
Odessa Astronoical Publications, vol. 8/ (05) 7 Table : The Neptunian oon syste (Neptune s radius,76 k). Nuber faily Moon naes R, k theoretical β M, kg Naiad 8 7 0.3597.9 0 7 Thalassa 50 075 0.358 3.5 0 7 0. 3 3 Despina 5 56 0.8039. 0 8 Galatea 6 953 0.76380. 0 8 5 Larissa 73 58 0.079 0..9 0 8 6 I S/00 N 05 00 0.96505 3 0.3 7 Proteus 7 67 0.335858 5.0 0 9 63.5 0.6 5 8.6 0.70 8 6 Triton 35 800.0000000.. 0 α=3.7 53. 783 060 3 060 5 II 70 08 5 65 3500 6 3775 0 β=0.6 9 Nereid 79 360.7 0.099068 0.5 9.6069 3. 0 9 77.8 0.5 0 3 Haliede 9 95 0.35886 0.3(3).0078 9.0 0 6 III Sao 9 05 0.85009 6.7 0 6 0.5 Laoedeia 33 38 0.979 5.8 0 6 3 5 Psaathe 37 3 65 0.85595 0.75.5 0 6 6 Neso 5 053.0000000..75853.7 0 7 α=3.0 w let us exaine the Saturnian oon syste: Drawing an analogy between acrocos and icrocos, in accordance with the planetary odel of the ato in which an electron strives to occupy the lowest orbit fro the allowed ones, it can be assued that a siilar phenoenon can be observed in acrocos as it was in the case of the β paraeter selection during assignent of the second of the Neptunian oons. It eans that the allowed orbits of a central body s oons are deterined on the sae ground. As can be seen, the oons of the Saturnian syste are divided into three s. The oons fro the 0 th to the 3 th for a sub located between the orbits of the first and second oons within the first. This can be called a faily or a sub of the first oon within the first. One of the criteria by which the oons were assigned to this, is the oons sizes given in the last colun of the table as it is not feasible to perfor any other assignent. It should be noted that the 9 th oon of the first faily within the third of the Saturnian syste satisfies the following condition: h h 6938 (see 3,,9, Table 5). Further let us consider the oon syste of Jupiter. Using the sae principles as before, we obtain data presented in Table 6. The distance for the first oon of the first of the Jupiter syste is deterined by relation h h, although it is less than the central 6 body s radius (see Table 6). The Uranian oon syste can be described with four s (see Table 7). In different sources, the solar and planetary paraeters vary significantly. Table 8 presents soe variations of those paraeters, as well as the obtained values of paraeters α and β.
7 Odessa Astronoical Publications, vol. 8/ (05) Table 5: The Saturnian oon syste (Saturn s radius 60,68 k). Nuber faily Moon nae R, thsd. k theoretical 0 S/009 S 7 0.095776 0,3 Pan 33 0.088697 0 3 Daphnis 36.5 0.737 0.0 7 5 Proetheus 39. 00 3 Enceladus 38. 0.990 99 5 Tethys 9.7 060 6 Telesto 9.7 0.35 0.36 I 7 3 Calypso 9.7 9 8 Dione 377. 8 9 3 Helene 377. 0.30898 0.3(3) 3 0 3 Polydeuces 377. Rhea 57. 0.368 0.38 58 5 85 0.6673 6 Titan.63. α=3.37. α=3.3 550 3 Hyperion 63.98 0.38775 0.6.007 66 Iapetus 3560.09 0.76998 0.39.08 36 3 7 0.36 0.99 II 6 0.76.6 5 5 Kiviuq 0787.8 0.8390780 0.7 3.3 6 6 Ijiraq 0835.5 0.888 7 6 Phoebe 856.073.. α=3. 0.5 0 β =.57 8 Paaliaq 669.37 0.590656 : 0.0 9 Albiorix 565.908 : 0.36 3 3 Bebhionn 6088.0 : 0.99 6 35 8 Skoll 666.089 0.669005 :.36 6 36 9 Siarnaq 736.7 0.689995 0 39 S/00 S 7 7870.709 0.795 :. 6 5 7 Farbauti 070.5 0.8097 : 3.78 5 57 III 30 Kari 079.78 0.83599 :.6 7 6 3 Loge 860.66 0.907 : 5.6 6 6 35 Fornjot 837.8.0 : 6.97; : 6.98 6 3950 90 3 5050 7980 70560 7083 5 0900 809 β =.57 6 63535 87 β = 33.869 β D (α 3., β.57, αβ 37.0)
Odessa Astronoical Publications, vol. 8/ (05) 73 Table 6: The oon syste of Jupiter (Jupiter s radius 7,9 k). Diaeter Nuber faily Moon nae Average radius, thsd. k theoretical 0.9 0.3 3.89 0. 3 57.67 0.38 I 8.70 0.67 ~0 Metis 7.69 0.70059 5 ~6 Adrastea 8.69 0.7955960 0.709 3 ~6 6 Aalthea 8.366.0.0 α=3. ~98 Thebe.87 0.7890 0.5 0.83 5 ~3630 Io.7 0.3989 0.6 9.6 6 ~3,6 3 Europa 67.0 0.3567 0.36.6 II 7 ~56, Ganyede 070. 0.5685575 0.56.6 5 53.8 0.8.09 8 ~80,6 6 Callisto 88.68.0.0 0.38 α=.8 β=. 96 6 β ф85 0.533 63 553 β и73 0.3309 9 8 3 Theisto 7309. 0.399 0.399 0.89 0 0 Leda 08.66 70 Hialia 385.86 0.50088 0.63 86 3 Elara 66.67 0.739336 3 36 Lysithea 688.9 5 S/000 J 35.6 5 S/003 J 6787.83 0.87 5 0.776 6 3 Carpo 68.85 0.7396097 0.89 7 Euporie 90.30 0. III 6 Thelxinoe 007.99 0.66 3 8 6 Ananke 0787.9 /каллисто=.0 33 7 Herippe 0898.75.0 6 3 8 Thyone 055.9 β к=086 β=0.86 К 6 38 S/003 J 0 07. /каллисто=.69 5 60 35 Pasiphae 73.76.00000000.0 /каллисто=.08 5 6 38 Care 873.3 /каллисто=.5 6 38 8 Sinope 3589.5 /каллисто=.53 65 9 Isonoe 360.9 /каллисто=.5 α=3. 66 5 Megaclite 080.9 0.8 67 S/003 J 3008.99.638 5900 IV 3 8900 750 5 7300 6 5600 β=.0 α=3. (α 3.0, β., αβ 3.05 ) β
7 Odessa Astronoical Publications, vol. 8/ (05) Table 7: The Uranian oon syste (the radius of Uranus is,800 k). R thsd. k Nuber faily Moon nae r/r β β Theoretical r/r 3.83 0.85. 0.895 9.6 3 0.396 9.75000 Cordelia 0.65006 53.7669 Ophelia 0.70355 0.65 7 66.097000 7 Portia 0.86975 8 69.97000 5 Rosalind 0.950353 0.935 9 7.800000 Cupid 0.978803 I 0 75.55000 6 Belinda 0.987553 76.0000 3 Perdita. α=.53. β=6.5 86.00000 Puck 0.7386 6.3 3 97.73000 Mab 0.67378 0.6 9.389950 Miranda 0.765 6.09 0.5 5 9.09930 3 Ariel 0.37335 6.53 0.33 6 66.98930 II Ubriel 0.563838 5.3 0.5 7 35.909790 5 Titania 0.770. 6.36 0.75 8 583.59630 6 Oberon. α=.99 7.637. 68 656 89 3 3 09 8 685 73 III 5 38 758 6 3067 369 β = 6.39 9 5.6700 Francisco 0.3996 6.59 0 78.70300 Caliban 0.36339 7.69 796.08700 3 Stephano 0. 80.67800 Trinculo 0.3 6.359 3 97.873000 Sycorax 0.56888 36.7000 Margaret 0.569 6.59 IV 5 580.5000 5 Prospero 0.795 6 639.657000 Setebos 0.869 6. 7 9879.088000 6 Ferdinand.. α=. 79 5.837 α=. (α.65, β 6.3, αβ 6.75) Table 8: Dynaic paraeters of the Sun and solar syste planets. Planetary The core Volue (V), naes teperature, T cub. I О I О * α β αβ Sun.35.5 0 7. 0 7 0.7 0.3.9 3. 38.5 Jupiter 0 5 0 3.3 5. 0 3 0,0 0.6 3.0. 3.05 Saturn.7 0 0 3 8.7 9.3 0 3 0, 0.7 3..57 37.0 Uranus.737 0 3 6.39 6.833 0 0,3 0..65 6.3 6.75 Neptune 7 0 3 6,5 6.58 0 0,6 0. 3. 0.6 3.98 Here I О is the reduced oent of inertia.
Odessa Astronoical Publications, vol. 8/ (05) 75 Table 9: The S paraeter values for the solar syste giants and the Sun. Planetary The core Volue (V) naes teperature, T cub. I 0 αβ S S~ Sun.35 0 7, 0 7 0.3 38.5 0.07977٠0 0 0.08٠0 0 Jupiter 5 0 3,3 0 0. 3.05 0.08399٠0 0 0.08٠0 0 Saturn.5 0 3 8,7 0 3 0. 37.0 0.08356٠0 0 0.08٠0 0 Uranus.5 0 3 6,833 0 0. 6.75 0.0835٠0 0 0.08٠0 0 Neptune. 0 3 6,5 0 0. 3.98 0.0830٠0 0 0.08٠0 0 The given values of the paraeter S indirectly sustain the planetary spacing rule. Having the values T, V and I O selected (fro Table 8), we see that the paraeter S, deterined by the following forula: V S, (8) T0 I takes on close values for planet giants and the Sun.. Conclusions Forally, α, in the units of the 3 rd oon, is the upper boundary of the first or the distance to the 6 th oon. Then, αβ is the distance to the 6 th oon within the next or the upper boundary of the second. Thus, a set of values α and β can be deterined fro forulae () (7) using two radii of the orbits of oons assigned to the given positions. Coparing these values with the values of T, V and I О in forula (8), the fittest paraeter values can be found. References. Bakulev V.M. The Titius-Bode law of planetary distances: new approach (arxiv:astro-ph/060369 006astro.ph..369B).. Cuntz Manfred: 0, Publications of the Astronoical Society of Japan, 6,., Article.73, 0 pp. 3. Griv E.: 006, European Planetary Science Congress. Berlin, Gerany, 8 Septeber 006, p... Hayes Wayne, Treaine Scott: 998, Icarus, 35, Issue, pp. 59-557. 5. Kotliarov I. The Titius-Bode Law Revisited But t Revived (arxiv:0806.353 008arXiv0806.353K). 6. Pankovic Vladan, Radakovic Aleksandar-Meda. A Close Correlation between Third Kepler Law and Titius-Bode Rule (arxiv:0903.73 009arXiv0903.73P). 7. Poveda A., Lara P.: 008, Revista Mexicana de Astronoía y Astrofísica,, 3-6 (http://www.astroscu.una.x/~raa/). 8. Zawisawski Z., Kosek W., Leliwa-Kopystyski J.: 000, A&AT, 9, Issue, 77-90.