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Planetary systems based on a quantum-like model

Planetary systems have their origin in the gravitational collapse of a cloud of gas and dust. Through a process of accretion, is formed a massive star and a disk of planetesimals orbiting the star. Using a formalism analogous to quantum mechanics (quantum-like model), the star-planetesimal system is described and the flow quantizing the gravitational field theoretical model parameters are obtained. Goodness of fit (chi-square) of the observed data with model quantum-like, to the solar system, satellites, exoplanets and protoplanetary disk around HL Tauri is determined. Shows that the radius, eccentricity, energy, angular momentum and orbital inclination of planetary objects formed take discrete values depending only on the mass star.

In quantum mechanics, Planck's constant is extremely small, h ∼ 10 −34 J.s and the ratio de Broglie wavelength is inversely proportional to momentum. For a massive object length de Broglie wave is very small, and consequently the wave behavior of a macroscopic object is undetectable, making quantum mechanics reduces to classical mechanics. Thus quantum mechanics is a theory developed to explain phenomena only a microscopic scale.
Planetary systems originate from a nebula which collapses to form a star and a disk of planetesimals orbiting the star. The trajectory of the planetesimals is determined by the principle of stationary action, δS = 0. However, planetesimal is disturbed by the presence of the other, oscillating around the classical trayectory; this causes for a closed and stable trayectory, the action is quantized; that is, the action is an integer multiple of a quantum of action, S = nh s (see Appendix A). The quantum of action, h s (which plays a role equivalent to Planck's constant h) is a free parameter to be determined and depends on the physical system in question. Considering the flux quantization of the gravitational field, the value of h s is determined. For a massive object: h s ≫ h, this allows to describe a macroscopic system with the formalism of quantum mechanics (quantum-like model). This means that the formalism of quantum mechanics is applicable to any system that quantize the action. One could use the Bohmian interpretation to give meaning to the formalism.
This implies that planetary systems (planets-Sun, satellites-planet, exoplanets-star, exoplanets-pulsar) must quantize not only the orbital radius as demonstrated, but also eccentricity, orbital inclination, angular momentum and energy. Through the goodness of fit (chi-square) shows that there is a very good agreement between the observed data and the quantum-like model, although the formulation is based only on the interaction between two particles.

II. FLUX QUANTIZATION OF THE GRAVITATIONAL FIELD
In a microscopic system, the electric potential energy between a proton and an electron, is given by, U e = −e 2 /4πǫ o , and an equivalent macroscopic system: the gravitational potential energy between a star and a planetesimal, is given by, U g = −GM s m/r. A relationship of proportionality between the electric potential energy and the quantum of action h (Planck's constant), is assumed: U g ∼ h, also between the gravitational potential energy and h s parameter: U g ∼ h s , which plays the same role of Planck's constant, in a macroscopic system. We can establish the relationship between these two energies, to obtain a dimensionless independent amount of r, and normalize the macroscopic parameter on a unit, the Planck constant: U g /U e ∼ s / = .
The flow of the gravitational field generated by the mass of a star, through a spherical gaussian surface, is given by: Φ g = 4πGM s and the electric field flux generated by a proton is: Φ e = e/ǫ o . If we take the relation between the intensity of the gravitational and electromagnetic force in terms of their flows, Φ = mΦ g /eΦ e , we have, Φ ∼ . It is considered Φ, as flow of the quantum of action per unit solid angle: Φ = /Ω (Ω acts as the proportionality constant), we obtain

III. ORBITAL DYNAMICS MODEL
In classical mechanics, the trajectory followed by a planetesimal, orbiting a star, is an ellipse with semi-major axis a and eccentricity ǫ. The average value of the distance planetesimal-star during a complete orbit, turns out to be r = a(1 + ǫ 2 /2). The total mechanical energy is E/m = −GM s /2a and the magnitude of the orbital angular momentum is L/m = GM s a(1 − ǫ 2 ). By conservation of angular momentum, in the accretion process of planetesimals, planets must be in the same plane θ = 0. The equation of motion for the ith planet orbiting a star is:r i = −G(M s +m i )/r 2 i +F i , where F i are small magnitudes, which contain the perturbative effects of all other objects on the ith planet, which is not a point particle; this causes the parameters which determine the orbit, vary periodically. The variation of the orbital kinetic energy, due to the variation of the eccentricity is: i )/r i and, due to the variation of the orbital inclination: In the quantum-like model, by substituting the Hamiltonian, H = − 2 s ∇ 2 /2m − GM s m/r (∇ 2 is the Laplacian in spherical coordinates) in the Schrödinger-like equation, we obtain: E n /m = −E s /n 2 , where E s = GM s /2a s and a s = ( s /m) 2 /GM s . Also, the average distance, r n,ℓ = a s /2 3n 2 − ℓ (ℓ + 1) ; the magnitude of the angular momentum, L ℓ /m = ℓ (ℓ + 1) s , and its orientation, θ ℓ,m ℓ = arccos m ℓ / ℓ (ℓ + 1) . The quantum numbers leading to orbit and where the probability is maximum, are n = 1, 2, . . ., ℓ = n − 1 and m ℓ = ±ℓ. The sign of m ℓ corresponds to normal and retrograde orbits, respectively. With the wave function: (where ρ = r/a s ), the probability of finding a planetesimal in space |ρψ n (ρ, θ, φ)| 2 is obtained (see Figure 1). If the planetesimal obeys the quantum-like model, depending on the distribution of matter and the process of accretion of planetesimals, a large object (planet, dwarf planet or asteroid) is formed on the mean value of the distance r t ≡ a s n(n + 1/2), ie, the object must orbit the star formed, following an elliptical trayectory with semi-major axis a t ≡ a s n 2 , eccentricity ǫ t ≡ 1/ √ n and inclination θ t ≡ arccos 1 − 1/n (because this is the most probable, and stable closed orbit); with a total energy, E t /m ≡ −E s /n 2 and magnitude of the angular momentum, L t /m ≡ n(n − 1) s . The orbital inclination and eccentricity of the formed objects changes cyclically over time [Lisiecki 2010], so the observed total energy E o /m = −GM s /2a o should be corrected: in this equation △ε corresponds to other effects, which generally lead to a loss of energy, such as planet-asteroids collisions and others.

IV. THE SOLAR SYSTEM
On the origins of the solar system, the formation of objects by the process of accretion of planetesimals can occur anywhere in the protoplanetary disk, but the stable closed orbits give rise to larger objects, according to the quantumlike model, this occurs in a t = ka ⊙ n 2 . Based on the uncertainty principle, △p△a t ≃ s , we have that △a t = ka ⊙ . For an average radius of ∼ 300 km, the icy moons and rocky asteroids in our solar system passing from a rounded potato to a sphere [Lineweaver 2010], for this reason, we selected objects with a diameter greater than Vesta ≥ 525.4 km (NASA JPL Small-Body Database), smaller objects are discarded because they can easily change orbit, due to collisions or internal dynamic processes (eg, Yarkovsky effect). Assuming that the scaling factor is k = 1, assigning a quantum number to each object and selecting the of greater diameter: (4) Mercury, (5) Venus, (6) Earth, (7) (1), and the quantum numbers. Once have formed interior objects orbital radius is maintained, while objects in the energy continuum (≥ 30 AU) can be easily moved (Kuiper belt). Jupiter is at the inflection point. Is obtained E o ⊙ /m = (14.65188 ± 0.00160) GJ/kg, χ 2 DoF = 0.00002, R 2 = 0.99956, and e% = 3.31. It is assumed that the percentage error is due mainly to the effect they have had the impact of asteroids on the inner solar system objects. In turn, the degree of fit, is evidence of the quantization of angular momentum, ie, the quantization of the orbital inclination and magnitude of the angular momentum in the period of formation of the solar system.
In Figure 4 shows the observed angular momentum, which depends on the eccentricity, L o /m = GM s a o (1 − ǫ 2 o ); the objects far away from the Sun have very high orbital inclinations, this generates a variation in the orbital eccentricity (1), which in turn produces an anomalous behavior in the angular momentum. Suppressing effect of orbital inclination on the eccentricity, we can obtain a corrected angular momentum: L c /m = GM s r o (1 − ǫ 2 c ), is obtained: o ⊙ /m = (0.76144 ± 0.00081) PJs/kg, χ 2 DoF = 0.03682, R 2 = 0.99992, and e% = 0.11. The Sun corresponds the quantum number n = 1, consequently, its angular momentum is zero and the probability distribution of matter is a sphere, see Figure 1(a), this could explain why the angular momentum of the Sun corresponds only ∼ 2% of the entire solar system. The quantum numbers n = 2, 3, . . ., have a probability distribution of matter in the form of concentric toroids, see Figure 1(b), which have different degrees of inclination and orbital velocity, the absence of planets in numbers quantum n = 2, 3, indicates that objects migrated outwards or planetesimals were captured by the Sun.

A. Orbital migration
The parameters obtained correspond to the current solar system, but the planets had a different distribution when they formed: it is assumed that the sequence of the inner planets was, (4) Mercury, (5) Venus, (6) Earth, (7) Mars, (8) Vesta and (9) Ceres. The Jovian planets has the sequence given by [Gomes et al 2005]. In this solar system objects representing at least 26 is considered that at the time of its formation, the same objects were present. Taking the maximum goodness of fit (maximum likelihood) for these conditions we obtain the sequence: (4) Mercury, (5) Venus, (6) Earth, (7)

B. Satellites
Is have selected the satellites of the planets of the solar system (Planetary Satellite Physical Parameters, JPL, NASA) with diameters ≥ 350 km. In order to compare with the solar system, has taken the ratio of the orbital radius and the scaling factor (a o /k). By the same procedure, used for the solar system, is obtained:

V. EXOPLANETS
This research has made using the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org [Wright et al 2011]. Have been selected systems with a single star with known mass and exoplanets, radio and orbital eccentricity known; assigning a quantum number and selecting the most massive, there are 219 exoplanets. In order to compare with the solar system, has taken the ratio of the orbital radius and the scaling factor (a o /k). The value obtained for the Bohr radius is: a o ⊙ = (3.03542 ± 0.00944) × 10 −2 AU, χ 2 DoF = 0.00108, R 2 = 0.99777, and e% = 3.70 (see Figure 6).
In classical mechanics, by conservation of angular momentum, the objects must be in the same plane, θ → 0 • (in the accretion disk), in the quantum-like model there is a quantization of angular momentum and its orientation, objects have higher orbital inclination between the closer you are to the star, this explains the elevated orbital inclinations found for exoplanets (θ → 90 e% = 0.51. The total energy shown large variations when the exoplanet is very close to the star (before continuous), this is due to variations in the eccentricity (△E ǫ(t,o) ) and the inclination (△E θ(t,o) ) of the orbit, is obtained E o ⊙ /m = (17, 57860 ± 0.36720) GJ/kg, χ 2 DoF = 0.24013, R 2 = 0.88301, and e% = 16.00.

A. Protoplanetary disk around HL Tauri
Considering a central protostellar mass for HL Tau given by k = 0.55 [Beckwith et al 1990, , has to be formed planets in a t = ka ⊙ n 2 . Based on uncalibrated image of the protoplanetary disk HL Tauri taken by ALMA (ESO / NAOJ / NRAO) where planets may be forming, several gaps are distinguished a t = (10. 06, 24.49, 46.94, 68.02) AU which correspond to n = (25, 39, 54, 65), respectively.

VI. CONCLUSIONS
The quantum-like model is applicable to any system that quantize the action, therefore, the formalism of quantum mechanics can be used in macroscopic systems where, the quantum of action, s ≫ . In this particular case, this occurs because the flow of gravitational field is quantized. The dynamics is determined only by the mass of the central object (as the case, star or planet) independently of the mass of the orbiting object. The probability distribution of planetesimals in space and the physical quantities (radius, eccentricity, energy, orbital angular momentum and its inclination) of the objects in the planetary system, take a certain value (quantized). Therefore, planets and exoplanets tend to be formed at a predetermined distance which depends on the mass of the star and the quantity of matter on the disc, obeying a t = ka ⊙ n 2 ; the orbit may be normal or retrograde. Similarly, the angular momentum and its orientation takes certain value, however, because the planets and exoplanets are not punctual objects, occur periodic variations, in the eccentricity and inclination orbital, causing a change in the value of the observed energy, equation (1). The obtained theoretical model can be used as a tool to parameterize and study the formation of planetary systems.
parentheses correspond to a constant which is denoted by h s . Note that the action is quantized, that is, S x (x o ) = S t (t o ) is given by n times a quantum of action h s , we will call macroscopic parameter, which must be determined for this physical system. The quantum of action h s when the planetesimal travels a x o = λ is h s = S x (λ), or when time elapses t o = τ is, h s = S t (τ ); these expressions are analogous to the relations of de Broglie and Planck, respectively: where, s = h s /2π called, reduced macroscopic parameter, k = 2π/λ and ω = 2π/τ . The perturbation of the trajectory of a planetesimal is caused by the presence of other planetesimals, causing it to move from its original path r o , a distance ξ = r − r o (which is very small, r ≃ r o ). Making a Taylor series expansion of the effective potential U (r) around of r o , its radial motion is given by: where p ξ = mξ is a transverse momentum with respect to the unperturbed motion. Consequently, the disturbance makes the planetesimal harmonically oscillate around the non-perturbed trajectory.
Based in Hamilton canonical equation,ṗ ξ = −∂H/∂ξ, we may obtain the wave equation, The corresponding solution has the form: As the path is periodic, ψ(0, 0) = ψ(x o , t o ) = 0 then: k → k n = nπ/λ, ω → ω n = nπ/τ being (n = 1, 2, . . .). As we can see, the boundary conditions have the effect of discretizing the wave number and frequency. The solution of the wave equation takes the form: ψ n (x, t) = C n exp [i(k n x − ω n t)]. However, substituting relations (A1) in ψ n (x, t) there is no correspondence between the wave speed and the speed of planetesimals. Planetesimals to represent using the wave equation, it is necessary to modulate the amplitude of the wave with a positive real function, A(x, t) = A(x − v o t), to limit the extent of the wave and cause the envelope to move with a group velocity, v o . Obtaining, in general: the exponential allows taking into account the phenomena related to the superposition of waves, while the coefficient corresponds to a distribution function (or probability density) to find the planetesimal in space: ̺ = |Ψ(r, t)| 2 . By integrating over all space the complex constant C can normalize to unity. The probability current is defined as J = ̺v o . Conservation of probability is given by the continuity equation: the probability density moves in space with the same velocity ∇S/m = v o and follow the path set of planetesimals. The Hamilton-Jacobi equation we add the term U Q = 1 2 k o ξ 2 which corresponds to the effect of the disturbance, this implies that the classical force is being affected by another force, which generates the quantization of the action of the physical system: F = −∇U − ∇U Q . The eikonal equation: ∇ 2 A + (2mU Q / 2 s )A = 0 allows us to define a relationship between U Q and A. Substituting this expression in equation (A4) and with (A3) we can construct an analogous equation to Schrödinger, which call Schrödinger-like: From the Hamilton-Jacobi equation (A4) we obtain the trajectory followed by the particles and from Schrödinger equation (A5) we obtain the distribution of particles in the space or the probability density, ρ(r, t).
In the star-planetesimal system planetesimals that quantized action giving rise to stationary waves (or stable trajectories) which store the energy of the disturbance. These waves interfere constructively giving rise to phenomena of resonance or stationary states (when the system with the resonance frequency is disturbed), are described by the eigenvalue equation Hϕ (r) = E n ϕ (r). Planetesimals not quantized action are represented by traveling waves transporting the disturbance energy, these waves end transferring its energy, so that the planetisimals assume the classical trajectory, which is described by (A4) where U Q = 0, ie, the formalism of classical mechanics. Therefore, we have a privileged system where states appear when U Q = 0, which can be described with the formalism of quantum mechanics.