A DESCRIPTION OF EXTRA-SOLAR PLANETARY ORBITS THROUGH A SCHRÖDINGER TYPE DIFFUSION EQUATION

ADVANCES IN SPACE DYNAMICS 4: CELESTIAL MECHANICS AND ASTRONAUTICS H. K. Kuga Editor 3- (4). Instituto Nacional de Pesquisas Espaciais INPE São José dos Campos SP Brazil. ISBN 85-7--9 A DESCRIPTION OF EXTRA-SOLAR PLANETARY ORBITS THROUGH A SCHRÖDINGER TYPE DIFFUSION EQUATION Marçal de Oliveira Neto Instituto de Química Universidade de Brasília Campus Universitário Asa Norte 79-9 - Brasilia - DF - E-mail: marcal@unb.br Liliane de Almeida Maia Departamento de Matemática Instituto de Ciências Exatas - Universidade de Brasília Campus Universitário - Asa Norte 79-9 - Brasília - DF - E-mail: lilimaia@unb.br Saulo Carneiro Instituto de Física Universidade Federal da Bahia 4-34 Salvador BA - E-mail: saulo@fis.ufba.br ABSTRACT The planetary orbits in our solar system have been successfully described by an approach similar to the Bohr atomic model [-3]. A Schrödinger-type diffusion equation has shown that quantum states associated to the asteroid belt are in very good agreement with the observed mean radii of the asteroids which perform the internal and external orbits in the belt. According to these results a model of quantum states condensation was proposed in order to infer the orbits of the planets in our solar system [4]. In this paper the use of the Schrödinger-type diffusion equation reproduces the observed mean distances for the extra-solar planets discovered up to now. A considerable number of these planets were found at the planetary mean distance of.5 AU from their stars which is the fundamental radius obtained by this approach. Planets with mean distances of. AU were also predicted by the model. A SCHRÖDINGER-TYPE EQUATION A Schödinger-type equation in a disk arises when we employ quantum mechanics to describe a planar system involving an attractive central field with a body of mass m moving around a body of mass M under an interaction potential V(r): 3

( g*) ψ ψ ψ + + + V ( r) ψ = E* ψ. () µ r r r r θ The term in parenthesis is ψ where is the Laplace operator in polar coordinates applied to a wave function ψ of the body of mass m and µ is the reduced mass mm/(m+m). The parameter E * stands for the energy of the system and g * is a constant with the dimension of the Planck constant. As long as the potential V is a function of the radial variable only we may look for a solution using separation of variables Replacing () into () and dividing by f ( r) Θ( θ ) we get: ψ ( r θ ) = f ( r) Θ( θ ). () Φ( θ ) Φ( θ ) θ = r f f ( r) r f µ + r r ( g*) r ( E * V ( r)). Since the term on the left depends only on θ and the term of the right depends on r both must be equal to a constant which we denote by l. Therefore we obtain two ordinary differential equations: Θ(θ) = l Θ(θ) θ and f f l µ + + ( * ( )) ( ) = + E V r * f r. r r r r ( g ) (4) (3) Equation (3) has a solution ilθ ( θ) e Θ =. If we assume the boundary conditions Θ ( ) = Θ(π ) we end up with l = 3 L i.e. l is an integer. Now we apply a rescaling in (4) by where β ρ = β r β > µe* µ GMm =. Furthermore we define γ = * ( g ) (g * ) and equation (4) becomes β 4

~ ~ f( f( l γ ~ + + + f( =. (5) 4 ρ ρ ρ ρ ρ ~ If we consider f ( = u( in (5) we obtain ρ γ ( l /4) u"( + + ( ) = 4 u ρ. (6) ρ ρ For this equation ρ = is a regular singular point and ρ = is an irregular singular point. Equation (6) is a confluent hypergeometric equation referred to as Whittaker's equation. This equation has a regular solution given by a hypergeometric series which converges if and only if γ = l ± ± k k = 3L By definition γ and assuming that the solution u ( satisfy the boundary condition lim u ( = and that u () is finite we must have ρ + n = γ = l + + k k = 3... ; l integer. So far we have the conditions l = 3... They can be regrouped in the form n n = γ = l + = 3 5 7 l + k 3 l + 5... l = k n For each pair n l the solutions u nl were obtained by using Maple V software. As long as we obtain f nl (r of the original equation (4). Once we know the collection u nl ( we can recover the solution ) of functions f nl (r) and Θ l (r) their products generate the wave equations ψ nl solutions of the Schrödinger-type equation. Now we define P n = π * l ( r) dr ψ nl ψ nlrdrdθ which is equivalent to P ( ) [ ( )] n l r dr = r rf r dr. The mean radius is obtained by which is equivalent to π * l l nl rn = rpn ( r) dr = ( ψ n rψ ) r drdθ l 5

r n l = [ r f ( r)] dr. (7) With these results we now return to the rescaling factor β and consider the mean radius of the planet Mercury as r 3. (If we used r all the set of results would be inconsistent with the observed mean planetary distances.) Under this assumption we obtain β r β ( ) ρ ρ e + ; u ( = c e ρ ρ and f ( r) = ce ( r ). Since [ rf ( r)] dr = we get From Maple V calculations and we obtain β =.387 c 4 3 = β. 7 r 3 = =.387 3 β 7 3 = 9.4397648 /(3 / ). Therefore we will consider in each case β = β n = 9.4397648(/n). RESULTS AND DISCUSSIONS Table I shows the theoretical planetary radii calculated from the states associated to the pairs (nl) from the fundamental radius until the value corresponding to Jupiter. These theoretical values were considered as reference states to those found for the observed distances in the investigated extra-solar systems. In Table II the studied stars the semi-major axis distances of the planets and respective eccentricities are presented as described in Schneider's Extra-Solar Catalog [5]. The stars which masses are not yet known were not taken into account. Therefore 9 planets among a total of discovered up to the present were investigated in this study. The table also shows the mean distances calculated from the average of the semi-major and semi-minor axis the later obtained from the observed eccentricities. In the last column it is presented the rescaled mean radii derived from the relation MR = M R where R is the average between the major and minor axis M is the star mass and R is the resulting rescaled 6

radius calculated assuming the star mass M as equal to the solar mass. This relation was obtained from our previous theoretical studies [46]. Tables III and IV show the frequency of coincidence between the rescaled extra-solar mean radii and our theoretical reference states as obtained from the solutions of the diffusion equation. It was considered an error of % and 5% below and above the reference states in the combined observed and theoretical numerical calculations. Figures I and II present the resulting graphics obtained from Tables III and IV showing the frequency in which the observed extra-solar mean radii coincide with the theoretical results. In these figures the observed semi-major axis were also shown for the purpose of comparing their frequencies with those calculated from the rescaled radii. The results show a very satisfactory agreement with the observed values except those with mean radius at.5 AU.5 AU and.5 AU. However theoretical calculations based on a Bohr-like model [4] can show that bodies may be found at.8 AU which is approximately the mean distance between.5 AU and.5 AU. Mean distances at.5 AU could not be predicted neither from a Bohrlike model nor from the solutions of the Schrödinger-type diffusion equation. It is interesting to note that the most frequent observed mean distances were those associated to the fundamental radius at.5 AU to the Earth radius at AU and to the belt asteroids from.49 AU to 3.37 AU. Table I - Theoretical mean radii r nl obtained by the model POSITION GRAPHIC NOTATION PAIRS (nl) CALCULATED MEAN RADIUS (AU) Fundamental FR (/) 55 Radius Mercury MER (3/) (3/) 387; 33 Venus VEN (5/) 83 Earth EAR (5/); (5/) 5; 995 Mars MAR (7/3) 54 Hungarian HUN (7/); (7/); 4; 99; 8 (7/) Asteroid BELT (9/); (9/); Belt (9/); (9/3) (9/4) Jupiter JUP (/33) to (3/4) 453 to 63 337; 33; 35; 88; 49 Table II - Extra-Solar Planets Data STAR SEMI_MAJ ECC MEAN RADIUS STAR MASS RESCALED RADIUS HD 49674 568 568 568 HD 767 49 49 5 545 HD 64 35 8 343 3773 HD 68746 65 8 648938 9 59775 HD 46375 4 4 4 HD 83443 38 8 37939 79 99789 HD 847 4 498 97936 7 33837 7

HD 7589 46 54 4596644 5 4864764 5 PEG 5 5 5 BD- 366 46 46 56 HD 6434 5 3 4654544 4654544 HD 873 4 3 499548 6 445998 HD 9458 45 45 5 475 Ups And 59 34 5898944 3 7667787 Ups And 83 8 83643 3 78836 Ups And 5 4 39693 3 3738974 Epsilon Eridani 33 68 9599969 8 367996873 HD 3859 3 7496 39 6636847 HD 3859 35 34 34544664 39 47335789 HD 48 69 4 6893379 93 57768 HD 79949 45 5 4497857 4 557653 55 Cnc 3 997544 3 3745 55 Cnc 6 6 5978668 3 65457 HD 8943 73 54 67773 5 75888 HD 8943 6 4 9963 5 644683 HD 54 3 3 386439 386439 HD 4783 969946 9 3365 HD 479 8 33 7469 93 888357 HD 374 585 58353385 9 53578 HD 374 95 4 8685983 9 5744445 HD 4753 6 5 684744 9 74674764 HD 33 88 48 8794983 79 69479933 rho CrB 3 8 99549 95 845766 HD 565 49 9 4794758 3 54887 G 777A 365 365 9 385 HD 384 4 48 384844 5 4484 HD 7783 43 9544444 7 354899 HD 77 7 4 6965533 98 68697 HD 77 97 45 38366 99 7949 HD 4 98 37 9455559 397485 HD 744 8 7988988 45858386 6 CygB 7 67 486933 495873 HD 7456 76 649 49888 5 55384 HD 7456 447 395 488533 5 4566498 HD 34987 78 5 76765876 5 8599667 HD 43 9 53 75875 6 6758837 HD 8874 7 59985 59985 HD 68988 7 4 765378 8478454 HD 669 5 3 463547 8 58966 HD 9994 3 86867333 35 73779 HD 6437 7 34 69574338 7 894454 8

Gliese 876 7 6334 3 65957 Gliese 876 3 95333 3 4449697 47 Uma 96 9554 3 58493 47 Uma 373 376557 3 38377 HD 66 85 347 835968 7 884467 HD 66 6 5787333 7 75848946 HD 7659 34 8 3353999 95 358693 HD 83 6 488735 8 838545388 HD 6983 83 34 79848567 4 787993 HD 965 5 8 45 695 HD 7356 66 34 6434394 65347 HD 4979 88 4 7877359 8 8459547 4 Her 5 3537 499865 79 966934 GJ 3 49 55 45646439 9 48785 HD 866 439 97 38596 9 7643365 HD 959 4 5 399445 47694 HD 9788 94 36 9848758 6 96996837 Tau Boo 46 8 469657 3 65535 Gl 86 46 994779 79 868546 HD 34 3 45 943983 344376 HD 5554 38 4 69953684 49694953 HD 98 3 43 97767365 3 85797574 HD 39 69 896934 98 85866498 HD 58 35 7 53363 53363 HD 885 6 9999989 99 989897 HD 789 3 43 3875949 87 77346 HD 697 9977389 958 7 Vir 43 4 45755 4535583 HD 65 6 54 4337847 5 5354489 HD 89744 88 7 75485 4 55999 HD 377 6 56849833 95 4397434 HD 68443 9 59 68564 773666 HD 68443 85 8 84673 8459784 HD 33636 6 39 566746 99 494787 HIP 75458 34 7 4853 5 9895879 HD 4937 5 4 45384997 45384997 HD 399 334 6 9885 3783355 HD 476 3 334 938596 8 38936488 HD 368 335 366 5399838 4 794959 HD 6 7 77 75937 7 49439 9

Table III Number of rescaled mean radii which coincide with those calculated by the model within an error of %. Radii Semi-Maj. FR 7 3 Mer 4 5 Ven 4 6 Ear 3 8 Mar 5 3 Hun 6 7 Belt 7 8 Jup 3 Total 59 5 8 6 4 8 6 4 % FR Mer Ven Ear Mar Hun Belt Jup Radii Semi-Maj. Figure I Frequency of rescaled mean radii and semi-major axis that coincide with those calculated by the model within an error of %. Table IV Number of rescaled mean radii which coincide with those calculated by the model within na error of 5%. Radii Semi-Maj. FR 4 Mer 4 8 Ven 5 8 Ear 6 3 Mar 6 6 Hun 8 8 Belt Jup 3 Total 7 69 5 5 5 5% FR Mer Ven Ear Mar Hun Belt Jup Radii Semi-Maj. Figure II Frequency of rescaled mean radii and semi-major axis that coincide with those calculated by the model within an error of 5%. For larger values of n a great number of states is found [6]. However for extra-solar systems no orbits were yet discovered beyond Jupiter's radius. In the case of our Solar System as pointed out by Nottale et al. [] to explain the circularity of the orbits a great number of states occupying the same region of space most of them with large eccentricities leads to a strong chaos and to the crossing of orbits which on large time scales could generate the condensation of states on the observed approximately circular orbits. Therefore for n > 9/ we will consider only the states with rotational symmetry that is with l =.

If we denote by the triple (n l r) the resulting numbers n l and the corresponding mean planetary radius r nl (in AU) then the states (/5.3) and (5/9.34) can be associated with the orbits of Jupiter and Saturn respectively. The states (7/.) and (9/5.) give mean values around the orbit of the asteroid Chiron distant 3.7 AU from the Sun. The states (/8.3) can be associated with the orbit of Uranus while the states (5/5.9) has a mean distance close to the orbital radii of the recently discovered asteroids 993 HA and 995 DW. The next state (7/3.) clearly stands for the orbit of Neptune and finally the states (3/39.9) can easily be associated with the orbit of Pluto. CONCLUSIONS Theoretical results for the extra-solar planetary systems show a great coincidence between the planet distances and the radii corresponding to the fundamental state to the Earth and to the Asteroid Belt after the performing of a rescaling of distances considering the different stellar masses. Some observed radii do not coincide with the radii obtained by the flat solutions of the Schrödinger-type equation. This could be due either to the tri-dimensional nature of the corresponding systems or to uncertainty in the present observations. In the case of our solar system the theoretical results show very good agreement with the exterior orbits when only states with rotational symmetry (l = ) are taken into account. REFERENCES M. Oliveira Neto Ciência e Cultura (Journal of the Brazilian Association for the Advancement of Science) 48 (996) 66. L. Nottale G. Schumacher and J. Gay A&A 33 (997) 8. 3 A. G. Agnese and R. Festa Phys. Lett. A 7 (997) 65. 4 M. Oliveira Neto and L.A. Maia Advances in Space Dynamics A. F. Bertachini Editor pp 456-47 (). 5 http://www.obspm.fr/encycl/catalog.html. 6 M. Oliveira Neto L. A. Maia and S. Carneiro astro-ph/5379.