Some peculiarities in the Asteroidal Belt

The analysis of the fine structure of the Asteroidal Belt evidenciates a group of asteroids next to the resonance 4/9 with Jupiter. In this group and in other groups associated to the Hirayama families there are indications that their orbital parameters can be represented by quantum numbers as defined here and in two of our previous works. Together with this the distribution of the eccentricities and inclinations of the orbital planes of short period comets and diverse type of asteroids indicates that they can be classified as objects with e > sin i and objects with e > sin i with a limit e = sin i which determinates geometrical properties of the orbits related with discrete states in the solar system. This study lets open the possibility of following studies in order to confirm the quantum characteristics of the Asteroidal Belt being these characteristics common to all the solar system and depending of the same fundamental constant of action per mass unit H ₀ = 1/2 ₀ × T ₀ (potential × time) because only a small part of all the available data in the Asteroid Belt is used here.     

Period distributions in pulsating and binary stars

We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA _c in stable and pulsating stars. This parameter isA _c=Mv _eff R _eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM ²/R _eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv _osc,R _osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA _c (energy x time) (period)^1/3 independently ifA _c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP _eff in a stable star.From evolving stellar models one has that P _effP _eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn _k=kn ₁ withk integers andn ₁=(P/P ₁)^1/3 withP ₁=P _eff1.22 hr. In these scales proportional toA _c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn ₁,n ₂,n ₃, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n _k) in RR Lyrae and Cephei are also associated with discrete levelsn _k.the suggested scenario is that the potential energies and the amounts of actionE _p(t), A_c(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ _F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.     

Dynamical conditions in planets and stars

With the available data in planets, stars and galaxies, it is studied the functions of angular momentaJ(M) and amounts of actionA _c(M) (associated to the non rotational terms in the kinetic energy). The results indicate that independently of how are these functionsJ(M),A _c(M) their ratioA _c/J remains a near invariant. It is independent also from the type of angular momenta: intrinsic spins of the bodies or the total angular (orbital) momenta of the bodies forming a system; for instance, the Solar System and the planets.The relationA _c(M) for the Solar System are analogous to these in the FGK stars of the main sequence, and the relationJ(M) (also for the Solar System) is analogous to the lower possible limit for binary stars.The different types of binary stars from the short period, detached systems to contactary systems, gives a range of functionsJ(M),A _c(M) that are the same that one can expect in stars with planetary systems. According to the detection limits given for planetary companions by Campbell, Walker and Yang (1988) (masses of less than 9 Jupiter masses and orbital periods of less than 50 years) we calculate the limits forJ(M) andA _c(M) This gives a lower limitA _c/J 1 associated to stars with planetary systems as 61 Cygni and to short period detached binaries. The upper limitA _c/J 16 correspond to planetary systems as the ours and probably to cataclysmic binaries. There are reasons to suspect that systems as the ours and in range 4 A _c/J 16 (with a lower limit analogous to contactary binaries as Algols and W Ursa Majoris) must be the most common type of planetary systems. The analogies with the functionsJ(M)A _c(M) for galaxies suggest cosmogonical conditions in the stellar formation.Independently of this, one can have boundary conditions for the Jacobi problem when applied to a collapsing cloud. Namely, from the initial stage (a molecular cloud) to the final stage (a formed stellar system: binary or planetary) the angular momenta and amounts of action decayed to 10^~4 the initial values, but in such a form thatA _c(t)/J(t) remains a near invariant.     

Angular momenta in the solar system

A statistical study of the orbital parameters of comets, asteroids and meteor streams shows that the vectors representing their angular momenta per mass unit (or the average angular momentum for meteor streams) are not arbitrarily distributed in the space: They are clustered around determinated values of angles . This synthesizes the eccentricities and inclinations of the orbital planes in a unique parameter adequated for the statistical purposes of the present work being defined by cos = cos (arc sin e) cos i.The discreteness of the obtained distribution N() and its relation with the components of the angular momenta per mass unit is analysed having this distribution common features for objects of different nature and located in different places in the solar potential well. Some hypotheses concerning to these effects are discussed.     

Local galactic field of forces and the interstellar matter

The density distributions of the two main components in interstellar hydrogen are calculated using 21 cm line data from the Berkeley Survey and the Pulkovo Survey. The narrow, dense component (state I of neutral hydrogen) has a Gaussianz-distribution with a scale-height of 50 pc in the local zones (the galactic disk). For the wide, tenuous component (hydrogen in state II) we postulate a distribution valid in the zones where such a material predominates (70 pc?z? 350 pc the galactic stratum) i.e., $$n_H \left( z \right) = n_H \left( 0 \right)exp \left( { - \left( {z/300{\text{ }}pc} \right)^{3/2} } \right).$$ Similar components are found in the dust distribution and in the available stellar data reaching sufficiently highz-altitudes. The scale-heights depend on the stellar type: the stratum in M III stars is considerably wider than in A stars (500–700 pc against 300 pc). The gas to dust ratio is approximately the same in both components: 0.66 atom cm^?3 mag^?1 kpc in the galactic plane. A third state of the gas is postulated associating it the observed free electron stratum at a scale-height of 660 pc (hydrogen fully ionized at high temperatures). The ratio between the observed dispersions in neutral hydrogen (thermal width plus turbulence) and the total dispersions corresponding to the real inner energies in the medium is obtained by a comparison with the dispersion distribution σ(z) of the different stellar types associated with the disk and the stratum $$\sigma ^2 \left( {total} \right) = \sigma ^2 \left( {21{\text{ cm line}}} \right) \cdot {\text{ }}Q^2 ,$$ from which we graphically obtainedQ ²=2.9 ± 0.3, although that number could be lower in the densest parts of the spiral arms. Its dependence on magnetic field and cosmic rays is analysed, indicating equipartition of the different energy components in the interstellar medium and consistency with the observed values of the magnetic field: i.e., fluctuations with an average of ～ 3 μG (associated with the disk) in a homogeneous background of ～ 1 μG (associated with the stratum). A minimum and maximumK _z-force are obtained assuming extreme conditions for the total density distribution (gas plus stars). TheK _z-force obtained from the interstellar gas in its different states using approximations of the Boltzmann equation is a reasonable intermediate case between maximum and minimumK _z. The mass density obtained in the galactic plane is 0.20±0.05M _⊙ pc^?3, and the results indicate that the galactic disk is somewhat narrower and denser than has usually been believed. The effects of wave-like distributions of matter in thez-coordinate are analysed in relation with theK _z-force, and comparisons with theoretical results are performed. A qualitative model for the galactic field of force is postulated together with a classification of the different zones of the Galaxy according to their observed ranges in velocity dispersions and the behaviour of the potential well at differentz-altitudes. The disk containing at least two-thirds of the total mass atz<100 pc, the stratum containing one-third or less of the total mass atz≤600–800 pc, and the halo at higherz-altitudes with a small fraction of such a mass which is difficult to evaluate.     

Hydrodynamic and turbulent motions in the galactic disk

Statistics in absorption 21-cm data show two main types of clouds at low galactic latitudes: dense small clouds, many of them with molecular cores, with dispersions σ≈1.5 km s⁻¹ and large clouds forming the fine features of the spiral arms (the shingle like features) with a dispersion range α≈3–4 km s⁻¹. Sizes and dispersions of both types of clouds are compatible with the Kolmogorov law of turbulence: σ∞d ^1/3. The large clouds forming the shingle-like features can be considered as the largest clouds of a Kolmogorov spectrum (the initial vortices), or as the hydrodynamic features with minimum sizes in the Galaxy. In order to define hydrodynamic motions in the same sense as given by Ogrodnikov (1965) we use here the tensorial form of the Helmholtz theorem to obtain an approximation for the hydrodynamic motions depending on distances and seen from the local standard of rest:V _r ∞r. The intermediate range of sizes between turbulent motions and hydrodynamic motions is 100<d<300 pc which is also the range of sizes of the large clouds forming the fine features of the spiral arms. A classification on of motions in the Galaxy is postulated: (a) a basic rotation motion given by an smooth unperturbed curveΘ _b (R) associated to the old disk population. (b) Systematic motions of the spiral arms. (c) Systematic motions in the fine structure of the arms. For scale sizes smaller than these fine features one has turbulent motions according to the Kolmogorov law. The densities and sizes of the turbulent clouds behave asn _H ∝d ⁻² in a range of sizes 7 pc<d<300 pc. The obtained gas densities of the clouds are confirmed with the dust densities from photometric studies. The conditions for gravitational binding of the clouds are analyzed. Factors as the geometry and the magnetic field within the clouds increases the critic densities for gravitational binding. When we consider these factors we find that the wide component clouds have densities below such a critical value. The narrow component clouds have densities similar or above the critical value; but the real fraction of collapsing clouds remains unknown as far as the factor of geometry and the inner magnetic field of each cloud are not determinated.     

Hydrodynamic and turbulent motions in the galactic disk,II

From a comparative study between stellar and gas data it is seen that turbulent and hydrodynamic motions in the Galaxy are common to both types of materials:

1. Galactic clusters have sizes and intrinsic dispersions compatible with the modified form of the Kolmogorov law seen in molecular clouds: undimensional velocities σ(km s^?1)=0.54d ^0.38 (pc). This indicates that ‘typic’ clusters were born from ‘typic’ dark clouds as these of the Lynds's catalogue (diametersd<10 pc, dispersions σ<1.5 km s^?1 hydrogen densitiesn _H>200 atom cm^?3). These clouds have mass enough to form galactic clusters (1000–3000M _⊙).
2. The cluster formation is related to the supersonic range of the Kolmogorov relationship σ(d>1 pc) while the AFGKM stars are related to the subsonic range of the same relationship σ(d<0.3 pc), the intermediate transition zone is probably related to OB stars and/or trapezia.
3. The effects of the magnetic fields in the clouds are also discussed. It seems to be that in the clouds the magnetic energy does not exceed the kinetic energy (proportional toσ ²(d)) and that this determinates the freezing criteria. The hypotheses introduced here can be checked with 21 cm Zeeman splitting.
4. Low-density globular clusters are also coherent with the Kolmogorov relationship. Some hypotheses about their origin and the type of clouds where they were born are discussed. This last part of the study lets open the possibility of further studies about evolution of globular clusters.

     

Angular momenta in binary systems

The correlations angular momentaL to massesM are studied for different types of spectroscopic binaries. The functionsL=AM ^b have the coefficientb with the values expected from a Keplerian mechanics, but the valuesA(q, T), A(q, a), A(q, v), associated tob=5/3, 3/2, and 2, respectively, are given (statistically speaking) by multiples or submultiples of discrete values of: the mass ratiosq, the semi-major axesa, periodsT, and velocitiesv of the reduced mass. This indicates the existence of a discrete unit of actionL=(1/2)×potential energy xperiod. Postulates about equivalent states of angular momenta for different orbital parameters are introduced, being this coherent with the analysis of the up-to-date data. Among other examples of the application of such equivalence postulates, we haveL(M) (W-type of the WUMa systems)L(M) (main group of the Algol binaries). The quantum units of action seen here are equivalent to those seen in the solar system in one of our previous works. From comparisons with galaxies and single stars, it is evidence that there is not an unique universal functionL=AM ^b, when the fine structure of the relation is analysed: each type of object has its own coefficients,A, b. It sems to be that there are an upper and a lower limit for all the possible functions. The upper limit isL=A _gM^5/3, withA _g1 associated to periodsT Hubble time, and the lower limit isL=GM ^2/c, with 1. The existence of the upper limit can be investigated with studies of pairs of galaxies, and the lower limit can be tested with analysis of single G, K, M stars. The quantical hypothesis introduced here can be checked definitely, when available larger samples of data with low errors, with similar quality as the selected list of almost 80 eclipsing binaries (mainly detached systems) analysed here.