1 Introduction

In 1530, Copernicus completed his major work, "On the Revolutions of the Celestial Spheres," in which he placed the sun at the center of the solar system. However, in his model, the stars were fixed and our sun and solar system were placed in the center of a closed circle. In 1584, Giordano Bruno published "Dell Infinito, universo e mondi" ("Of Infinity, the Universe, and the World"). Without the aid of a telescope, Bruno realized that "there are innumerable suns and an infinite number of planets which circle around their suns as our seven planets circle around our Sun." However, according to Bruno, we are unable to see these planets and suns "because of their great distance or small mass." Bruno also relied on the ancient Greek model of the atom to advance his planetary theory. According to Bruno, the stars and planets were like atoms and moved around in the same type of void and empty space that existed between atoms.

In the early 1900s three hundred years after Giordano Bruno was tortured and burned alive by the Catholic Church, a number of critical discoveries relating to the intrinsic discontinuous nature of electromagnetic radiation emitted by the hydrogen atom are responsible for key advances in atomic physics. In 1913, Niels Bohr (Bohr, 1913) formulated a groundbreaking planetary model of the hydrogen atom. By imposing the condition that the angular momentum (L) of the electron must be expressed as an integral multiple of Planck's constant (h), he basically created a model of the atom based on planetary orbits. In the resulting equation, L = nh / 2 π, the integer n represents a label assigned to each discrete electronic orbit starting with n = 1 for the fundamental state.

A decade later, Louis De Broglie (De Broglie, 1925) established the hypothesis of wave-particle duality as a general property of microscopic objects. The concept of matter waves was further extended by Erwin Schrödinger (Schrödinger, 1926), who presented an equation that allows the propagation of the wave function representing a given quantum system to be deduced from the Hamiltonian of the corresponding classical system. This fundamental equation provided the foundation for further theoretical investigations of the electronic structure of the hydrogen atom. Furthermore, this equation has been successfully applied to polyelectronic atoms and has played an essential role in the development of theories relating to the nature of chemical bonds.

Exploring in a direction opposite to that taken by Bohr, one might wonder whether atomic models relating to the universe of the small could be effectively applied to the large universe of planetary orbits. In this context, atomic-like models have been employed to explain planetary orbits on the basis of quantum theories related to the electronic structure of the hydrogen atom (Oliveira Neto, 1996, 2006; Oliveira Neto et al., 2004), whereas molecular quantum chemistry procedures have been used with some success to describe planetary orbits in binary stars (Oliveira Neto, 2007). Moreover, a number of other authors have adopted quite different physical approaches in theoretical investigations aimed toward the same goal (Nottale, 1993, 1996; Nottale et al., 1997, 2000; Agnese & Festa, 1997; Giné, 2007; Scardigli, 2007). Some of the procedures adopted in these studies have generated many empty orbits or have not been able to predict the orbits of a significant number of bodies actually observed in the solar system. Nevertheless, these investigations have revealed that it is possible for planetary orbits to be described using the principles of quantum mechanics.

However, one feature of the quantum atomic theory is electron donation. If, as first argued by Bruno, suns behave like atoms, then planets should also be shared between suns. What this suggests is that planets may be expelled from different solar systems, only to be captured by others (Joseph 2009; Joseph and Schild 2010). Once these "rogue" planets (donated electrons) become a member of a new solar system, it can be predicted they would assume orbits dictated by quantum mechanics. However, even if we accept the traditional accretion model of planetary and solar system formation, we should be able to make predictions of orbital dynamics based on quantum mechanics. As detailed in this paper, molecular quantum mechanics used to describe the electronic distribution around the nucleus of the molecular ion H₂⁺, can in fact predict the patterns of protoplanetary disks observed around binary system stars.

2 A Bohr Atomic-like Model Applied to Planetary Orbits As the majority of planets exhibit very small eccentricity the Newtonian law for the orbital velocity (v) of a planet in circular orbit around the Sun, v² = GM_s/ r , may be applied. Application of a Bohr-like "quantization" rule to the orbital angular momentum of the planet affords the following expression for the mean planetary radius (r) (Oliveira Neto, 1996; Oliveira Neto, 2006):

Here, n is an integer, G is the Newton gravitational constant, M_s is the solar mass, m_p is the average mass of the planets of the solar system and g* represents a "quantum of action" and takes the role of a re-scaled Planck's constant. Using this approach, the orbital velocity is then given by:

from which the orbital period (T) for a planet can be determined as:

or

Now, if n_i and n_j are integers corresponding to any two orbits, then:

and showing that the ratio between the mean radii of any two successive orbits is given by:

Similarly, the orbital periods of the two orbits would be:

from which:

Equation (5) and Equation (6) suggests the following general principle: the ratio between mean planetary distances corresponds to that of the squares of the integers associated with the respective orbits, whereas the ratio of their orbital periods relates to that of the cubes of the two integers. From this it can be deduced that the recurrence expressions for successive orbits n and n_i+1 is:

and

The application of Eq.7 and Eq.8 can be illustrated by considering the mean planetary radii and orbital periods, in appropriate astronomical units, within the solar system. The semi-major axis, which is the same as the mean planetary distance to the Sun, is expressed in terms of the mean distance from the Earth to the Sun (designated as one astronomic unit or AU). In astronomical units, the orbital period of the Earth (one year) defines the unit of time. Mercury is the closest planet to the Sun (hence its orbit corresponds to n = 1), and its observed mean radius (r1) and orbital period (T₁) are 0.387 AU and 0.241 years, respectively. Based on these values, the other mean planetary radii and orbital periods can be calculated from Eq.7 and Eq.8 by setting n in the range1£ n £13 (Table 1). Regarding planetary orbits where n = 2 and n = 3, the sum of the squares of n and a second ad hoc integer m, taking values from 0 to n, must be considered using expressions analogous to those of Eq.7 and Eq.8 namely:

and

This procedure may be illustrated by reference to the series of orbits with n = 2. Starting from the state n = 2, m = 2, which is associated with the orbit of Mars, the state n = 2, m = 1 corresponds to Earth's orbit and n = 2, m = 0 corresponds to that of Venus. The respective mean planetary radii and orbital periods are given by the following calculations:

in which the values r_2,2 = 1.548 and T_2,2 = 1.928 correspond to the parameters for Mars. It is worth noting that the splitting of states associated with n equal to 2 or 3 is analogous to the spectral series derived from modern atomic theory. Moreover, this splitting of states occurs in the region corresponding to the distance of Jupiter from the Sun (n = 4) and may be linked with the unusual characteristic of this planet that, together with its innumerable rings and satellites, almost constitutes a mini-solar system in its own right.

Regarding the terrestrial and the gas giant planets, as well as the dwarf planets Pluto, Makemake and Eris, the theoretical mean radii and orbital periods predicted by this model are in reasonable agreement with the observed values (Johnston's Archive; Space and Astronomy, 2010).

Furthermore, there is significant agreement between the theoretical and observed results (Table 1) regarding the positions of some asteroids found in the solar system. The model predicts, for example, the orbits of the inner (n = 3; m = 0; HIL) and outer (n = 3; m = 1; HOL) limits of the Hungaria asteroids at mean observed radii between 1.780 and 2.000 AU. The asteroids Vesta (n = 3; m = 2) and Camilla (n = 3; m = 3) are correctly located in the inner (2.361 AU) and outer (3.477 AU) rings of the main asteroid belt, which lies between the orbits of Mars and Jupiter and contains approximately 2000 objects orbiting the Sun. The asteroid Chiron, a Centaur object, is positioned between the orbits of Saturn and Uranus at an observed mean radius of 13.698 AU. Moreover, the calculated mean radius of 24.768 AU is associated with the recently discovered asteroids, also Centaur bodies, named Nessus and Hylonome, whose mean distances are 24.617 and 25.031 AU from the Sun, respectively. Additionally, this model predicts the orbit of trans-Neptunian objects in the region of space where the Plutoids are found, including that of the asteroid 1999 DE9 (n = 12) at an observed mean radius of 55.455 AU, a value that accords very well with the theoretical result of 55.728 AU (Johnston's Archive; Space and Astronomy, 2010).

Kepler's third law of planetary motion, as determined from classical mechanics, states that r³ /T² = GM_s/4π . Therefore, the theoretical mean planetary radii and the orbital periods given by Eq.1 and Eq.4, respectively, are in full compliance with the expression for this law. The observed and theoretical orbital velocities can be derived from the mean radii and periods in astronomic units as displayed in Table 1 by applying Eq.3. Using this unit system, Kepler's third law simplifies to T = r^{3/ 2} , and substitution of this expression into Eq.3 gives vr^{1/ 2} = 2π. For both theoretical and observed values, the square roots of the mean planetary radii multiplied by the velocities are approximately equal to 2π , in accordance with Kepler's third law.

3 A Schrödinger-type Diffusion Equation Applied to Planetary Orbits

The mean distance of the electron from the nucleus of a hydrogen atom can be evaluated from the wave function solutions of the Schrödinger equation. Similarly, the mean planetary radii may be associated with functions obtained from the solutions of a Schrödinger-type diffusion equation (Oliveira Neto et al., 2004). The basis of this hypothesis is that this type of equation can describe a classical diffusion process in the formation of the solar system, where Planck's constant plays the role of a diffusion coefficient. An appropriate equation can be developed by considering the flatness of the solar system in terms of the original disk from which it originated. Detailed consideration of this equation (Oliveira Neto et al., 2004) furnished nl y solutions in which:

and the mean radii are represented as:

Equation (11) can then be related to the probabilities of finding mass peaks in a given region of space around the Sun. Similar to the previous Bohr-like procedure, this model requires just one initial parameter, i.e., the mean planetary radius of Mercury, to predict the remaining radii (Oliveira Neto et al., 2004). A "fundamental state", associated with the pair (n=1/2, l = 0) and with a corresponding mean radius of 0.055 AU, is predicted by the procedure (Table 2). Interestingly, many planets in exosolar systems have been discovered in this position, including the exosolar planets called "hot Jupiters" (Schneider, 2010). Predictions of a similar "fundamental radius" have also emerged from the investigations of other authors (Nottale, 1996; Agnese & Festa, 1997).

The theoretical results obtained from the Schrödinger-type model for the terrestrial planets and asteroid belts are displayed in Table 2 and show good agreement with those derived from observations (cf. Table 1). The lower mean radius of 2.488 AU for the state (9/2, 4) is associated with the mass peak of the asteroid Vesta that delimits the inner ring of the main asteroid belt. The sets (9/2, 0) and (9/2, 1) correspond to the mass peak of the asteroid Camilla which, in turn, delimits the outer ring of this belt. The observed mean radii for the dwarf planet Ceres (2.766 AU) and the asteroid Pallas (2.773 AU), both lying between the inner and outer rings of the main asteroid belt, can be associated with the state (9/2, 3) and a theoretical radius of 2.875 AU. The fourth largest body in this region, the asteroid Hygiea, revolves in an orbit around the Sun at an observed distance of 3.139 AU, in good agreement with the theoretical mass peak of 3.151 AU associated with state (9/2, 2).

For values of n > 9/2, a very large number of states is found (Oliveira Neto et al., 2004). As pointed out by Nottale et al. when explaining the circularity of orbits in the solar system, a large number of bodies orbiting in the same region of space and presenting significant eccentricities leads to considerable chaos (Nottale et al., 1997). The crossing of these orbits over a long period of time gives rise to a condensation of states of the observed approximately circular orbits. Table 3 shows the mass peak results for n > 9/2 up to n = 33/2 for states that present rotational symmetry, i.e., when l =0. The theoretical results in Tables 2 and 3 are in very satisfactory agreement with most of the observed mean distances for the astronomical objects found in the solar system. In certain circumstances, experimental observations correspond to the average value of the mass peak positions associated with two states determined from Eq.11. The mass peak corresponding to the Centaur asteroid Chiron orbiting around the Sun between Saturn and Uranus at 13.720 AU is given by the mean values of the mass peaks corresponding to the sets (17/2, 0, 11.997) and (19/2, 0, 14.982) within reasonable accuracy, as shown in Table 3. A further example of this phenomenon is provided by the recently discovered trans-Neptunian dwarf planet Haumea, a Plutoid revolving around the Sun at 43.335 AU, close to the mass peak of Makemake. At approximately the same mean radius value, two trans-Neptunian objects, 20000 Varuna at 42.908 AU and 50000 Quaoar at 43.407 AU, have been detected (Johnston's Archive; Space and Astronomy, 2010). The stable orbits of these bodies have mean radii at an average distance between the mass peak of Pluto at 39.861 AU (corresponding to the state 31/2, 0) and that of Makemake at 45.168 AU. This fact may be associated with the rotational movement of the celestial objects, which was not considered in the resolution of the diffusion equation.

4 A Theoretical Quantum-like Model for Binary Star Systems

Considering the quantum-like models used here to describe planetary orbits, it is reasonable to enquire whether binary star systems, and the formation of protoplanetary disks around them, might also be treated using molecular quantum mechanical methods. In the LCAO-MO (linear combination of atomic orbitals to yield molecular orbitals) procedure of molecular quantum mechanics, the ground–state electronic wave function for the ion H₂⁺is represented as the normalized 1s atomic orbital of hydrogen centered on the two protons (Pauling & Wilson, 1935). Solutions of the Schrödinger equation for this electronic system predict that the equilibrium distance for the ground state of H₂⁺is 1.32 Å, whereas experimental results indicate a value of 1.06 Å. This theoretical estimate can be improved, however, by elaboration of the calculations using more advanced and sophisticated procedures (Levine, 1991). Based on this model for the ground state of H2+, it may be concluded that the stability found in the diatomic system is essentially due to a reduction in the electronic energy as a consequence of the shared electron interacting with two nuclei rather than one.

We have used a similar approach to estimate the average separation distances between double star companions in the recently discovered HD188753 Cygni triple star system (Oliveira Neto, 2007). On the basis of three-dimensional 1s wave functions, and taking a0 = 0.387 AU, i.e., the mean planetary radius of Mercury, the variational method predicts an "equilibrium distance" for the two stars of around 1.0 AU. This compares to the observed separation of 0.66 AU.

Interestingly, recent observational research has aimed at interpreting the geometry of protoplanetary disks in relation to the separation distances of the binary star systems in which they form. By studying the spectral energy distributions and typical temperatures (in the range 100-200 K) of dust disks around 69 main-sequence A3-F8 binary star systems, Trilling et al. were able to derive typical fractional luminosities. Infrared excesses were observed in 22 of the systems studied, and these excesses were interpreted as thermal emissions from dust produced by collisions in planetesimal belts. In 12 of these systems, dust distances were much larger than the binary separation of the system, implying the existence of circumbinary debris disks with typical separation distances of < 3.0 AU, based on dynamical stability arguments. In contrast, seven of the systems had dust distances that were much smaller than the binary separation of the system, indicating the existence of circumstellar debris disks with separation distances of > 50.0 AU. In the remaining three systems, the dust distances were similar to the binary separations, implying that their locations were somewhat unstable (Trilling et al., 2007). The geometric arrangement of the protoplanetary disks around binary stars found by these authors is displayed in Fig. 1.

There are clear similarities between the chemical bonding aspects obtained from molecular quantum mechanics as applied to the H₂⁺system and the general features of protoplanetary disk formation around binary stars, particularly those relating to the separation distance between binary stars. Thus, the circumbinary debris disks observed at separation distances of less than 3.0 AU are analogous to the electron cloud surrounding the two nuclei at the equilibrium distance in the H₂⁺molecular system.

Correspondingly, circumstellar debris disks observed for binary separations of greater than 50.0 AU are similar to the physical aspects of H₂⁺ion dissociation, resulting in one nucleus separating from the electronic cloud surrounding the other nucleus and ultimately producing a hydrogen atom and a proton. Additionally, binary star systems that do not present either circumbinary or circumstellar debris disks exhibit unstable electronic cloud geometry. This arrangement is analogous to that found when the two nuclei of H₂⁺move away from the equilibrium distance of the molecular ion.

5 Conclusions

The calculated results presented in Table 1 do not depend on the rescaling Planck's constant or on individual planetary masses. Nevertheless, the theoretical values are generally in close agreement with those measured for terrestrial planets, such as Venus, Earth and Mars and even for the giant Jovian planets, such as Neptune, Saturn and Uranus. The largest discrepancy is found for the mean planetary radius of Jupiter, for which the calculated value of 6.192 AU differs by approximately 19% from the observed value (5.203 AU). This deviation may be attributed to the fact that Jupiter contributes around 70% of the total planetary mass of the solar system, resulting in a strong gravitational effect between this planet and the Sun. It appears, therefore, that a Bohr-like atomic model can predict the main mass peaks around the Sun using only a single integer, n. However, the introduction of a secondary "quantum number" l, resulting from the application of a Schrödinger-type equation, extends the model to produce finer detail.

The theoretical mean distance calculations reported in the present study could also apply to exosolar systems whose stars have solar masses very different than those of the Sun. Although the proposed theoretical model for the arrangement of protoplanetary disk formation around binary stars does not take into account the attractive gravitational potential between double stars (Oliveira Neto, 2007), it would appear that the theoretical mathematical framework describing the formation of a protoplanetary disk in binary stars would be analogous to that used to obtain the electron distribution functions for the molecular ion H₂⁺.

The line of research presented in this work was supported by the foundations of quantum theory to describe the electronic orbits of hydrogen atom and to the study of the chemical bond nature of the molecular ion H2+. Quantum chemistry has advanced significatively in applying these concepts to the whole set of known atoms as well as to the theoretical investigation of the electronic structure of a huge number of molecules.

Recent investigations in astrophysics based on observational data have led some scientists to conclude that the planet formation event appeared to be more consistent with collisional fragmentation than collisional accumulation (Joseph and Schild, 2010). The reasoning is that when rocky material collides, it fragments and does not stick together and grow larger. Therefore, it is not reasonable to assume planets grow larger by collisional processes, but grow smaller.

This theory is analogous to electron transfer involving collisions between chemical species and the study of such processes is one of the fields of quantum chemistry. These discoveries in astrophysics have led to the formulation of alternative models for planetary formation.

From the knowledge that the parent star loses mass and consequently gravity in a dying solar system, Rhawn Joseph proposed the theory that rogue planets, i.e. outer planets will be ejected from the solar system when the parent star becomes a red giant prior to supernova (Joseph, 2009). New solar system in formation may then capture these rogue planets and new planetary objects may be formed through collisions.

Also against the model of collisional accumulation or accretion, Rudy Schild and Carl Gibson proposed the theory of primordial planets, by assuming that stars should form from primordial gas rogue planets rather than planets forming from stars (Schild and Gibson, 2010). The recent discoveries of the super-Jupiter planets supports this theory. Additional support for the Joseph-Schild-Gibson model, is the evidence that in the early stages of solar system formation planets crashed into one another, as for example, the formation of the Moon through the collision between Earth and a Mars-sized planet about 4.2 billion years ago (Joseph and Schild 2010).

The resulting conclusions from the theoretical approaches reported above to describe planetary systems may then to provide relevant clues about mass transfer by applications of quantum mechanical procedures to the scenario of the theory of collisional fragmentation and planet formation.

Acknowledgements

This author wish to express his thanks to Professor Kleber C. Mundim (Instituto de Química-UnB, Brazil) and Adão L. Bezerra (Universidade de Tocantins, Brazil) for drawing his attention to the publication of Trilling et al. and to Dr. Darly Henriques da Silva (Ministério da Ciência e Tecnologia, Brazil) for reading this manuscript and providing helpful suggestions.