A hypothetical estimated mass of the muon neutrino and the mass of the tau neutrino

Using the see-saw mechanism and estimation of the hypothetical mass of the electron neutrinom _e we find the hypothetical mass of muon neutrinom _µ and hypothetical mass of the tau neutrinom .     

Non-zero rest mass neutrinos and the cosmological constant

Recently it was pointed out that a non-zero cosmological constant can play a role in the formation of neutrino halos only in the case of neutrinos of very low rest mass (m _v <-0.1eV). However, phase-space considerations would requirem _v >50 eV if neutrinos dominate the missing mass in halos of large spiral galaxies and moreoverm _v >200 eV is implied in the case of dwarf spheroidals. These larger neutrino masses would be in conflict with observed constraints on the age of the Universe unless a cosmological constant is invoked.     

Some special restricted four-body problems—II. From Caledonia to Copenhagen

Special analytical solutions are determined for restricted, coplanar, four-body equal mass problems, including the Caledonian problem, where the masses M_i = M for i = 1,2,3,4. Most of these solutions are shown to reduce to the Lagrange solutions of the Copenhagen problem of three bodies by reducing two of the masses (m_i = m for i = 1,2) in the four-body equal mass problem to zero while maintaining their equality of mass. In so doing, families of special solutions to the four-body problem are shown to exist for any value of the mass ratio μ = m/M.     

One example of mass exchange in case AB in system 5m ⊙+4m ⊙

Evolution of a binary system with masses of 5m and 4m , respectively, and with orbital period of 1.41 days is studied by means of non-stationary model calculations under assumptions of conservation of total mass and total orbital angular momentum of the system. As a result of mass exchange between the components we obtain a binary with masses of 8.46 and 0.54m . Physical parameters of the final product indicate possible connection with shell stars.It is also pointed out that the new secondary component can become rotationally unstable soon after the end of mass exchange.     

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

The global flow of the parabolic restricted three-body problem

We have two mass points of equal masses m ₁=m ₂ > 0 moving under Newton’s law of attraction in a non-collision parabolic orbit while their center of mass is at rest. We consider a third mass point, of mass m ₃=0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their center of mass. Since m ₃=0, the motion of m ₁ and m ₂ is not affected by the third and from the symmetry of the motion it is clear that m ₃ will remain on the line L. The parabolic restricted three-body problem describes the motion of m ₃. Our main result is the characterization of the global flow of this problem.     

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.     

Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m₁>m₂, the system HD82943 for the new masses, for the case m₁=m₂ and the same system HD82943 for the values of the masses m₁<m₂ given in previous work. In the case m₁≥ m₂ there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m₁<m₂ all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.     

The restricted planar isosceles three-body problem with non-negative energy

We consider a restricted three-body problem consisting of two positive equal masses m ₁ = m ₂ moving, under the mutual gravitational attraction, in a collision orbit and a third infinitesimal mass m ₃ moving in the plane P perpendicular to the line joining m ₁ and m ₂. The plane P is assumed to pass through the center of mass of m ₁ and m ₂. Since the motion of m ₁ and m ₂ is not affected by m ₃, from the symmetry of the configuration it is clear that m ₃ remains in the plane P and the three masses are at the vertices of an isosceles triangle for all time. The restricted planar isosceles three-body problem describes the motion of m ₃ when its angular momentum is different from zero and the motion of m ₁ and m ₂ is not periodic. Our main result is the characterization of the global flow of this problem.     

Does the proton-to-electron mass ratio μ = m p /m e vary in the course of cosmological evolution?

The possible cosmological variation of the proton-to-electron mass ratio μ = m _p /m _e was estimated by measuring the H₂ wavelengths in the high-resolution spectrum of the quasar Q 0347-382. Our analysis yielded an estimate for the possible deviation ofμ value in the past, 10 Gyr ago: for the unweighted valueΔ μ / μ = (3.0±2.4)×10^-5; for the weightedvalueΔ μ / μ = (5.02±1.82)×10^-5.Since the significance of the both results does not exceed3σ, further observations are needed to increase the statistical significance. In any case, this result may be considered as the most stringent estimate on an upper limit of a possible variation of μ (95% C.L.):|Δ μ / μ| < 8× 10^-5 .This value serves as an effective tool for selection of models determining a relation between possible cosmological deviations of the fine-structure constant α and the elementary particle masses (m_p, m_e, etc.). This revised version was published online in July 2006 with corrections to the Cover Date.     

Equations of motion of the restricted problem of (2 + 2) bodies when primaries are magnetic dipoles and minor bodies are taken as electric dipoles

In this problem of the restricted (2 + 2) bodies we have considered two magnetic dipoles of masses M ₁ and M ₂(M ₁ > M ₂) moving in circular Keplarian orbit about their centre of mass. Two minor bodies of masses m ₁, m ₂(m _j< M ₂) are taken as electric dipoles in the field of rotating magnetic dipoles. These minor bodies interact with each other but do not perturb the primaries.We have found equations of motions which differ from that of Goudas and Petsagouraki's (1985).     

Period-mass ratio relations for eclipsing binaries with periods not exceeding 5 days

The dependence of mass ratio of eclipsing binary systems on period has been studied. It has been found that semi-detached systems obey the relationm ₂/m ₁=0.64–0.079P orm ₂/m ₁=0.29–0.028P depending upon whether the mass ratio is >0.3 or <0.3. The mass ratio of deteched systems generally appears to be independent of the period.     

Robe’s problem: its extension to 2+2 bodies

In the problem of 2+2 bodies in the Robe’s setup, one of the primaries of mass m^*₁m^{*}_{1} is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ ₁. The second primary is a mass point m ₂ outside the shell. The third and the fourth bodies (of mass m ₃ and m ₄ respectively) are small solid spheres of density ρ ₃ and ρ ₄ respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal. We assume m ₂ is describing a circle around m^*₁m^{*}_{1}. The masses m ₃ and m ₄ mutually attract each other, do not influence the motion of m^*₁m^{*}_{1} and m ₂ but are influenced by them. We also assume masses m ₃ and m ₄ are moving in the plane of motion of mass m ₂. In the paper, the equations of motion, equilibrium solutions, linear stability of m ₃ and m ₄ are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are unstable for all values of the mass parameters μ,μ ₃,μ ₄. There exist an infinite number of non collinear equilibrium solutions each for m ₃ and m ₄, lying on circles of radii λ,λ′ respectively (if the densities of m ₃ and m ₄ are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ,μ ₃,μ ₄, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon.     

Residual mass from atmospheric ablation of small meteoroids

Numerical solutions of the equations of meteor ablation in the Earth's atmosphere have been obtained using a variable step size Runge-Kutta technique in order to determine the size of the residual mass resulting from atmospheric flight. The equations used include effects of meteoroid heat capacity and thermal radiation, and a realistic atmospheric density profile. Results were obtained for initial masses in the range 10^?7–10^?2 g, and for initial velocities less than 24 km s^?1 (results indicated no appreciable residual mass for meteors with velocities above 24 km s^?1 in this mass range). The following function has been obtained to provide the logarithm of the ratio of the residual mass following atmospheric ablation to the original preatmospheric mass

$\text{log r = 4.7 ?0.33v}{}_{\mathrm{\infty }}\text{?0.013v}{}_{\mathrm{\infty }}{}^2\text{+ 1.2 log m}{}_{\mathrm{\infty }}\text{+ 0.08 log}{}^2\text{m}{}_{\mathrm{\infty }}\text{?0.083v}{}_{\mathrm{\infty }}\text{log m}{}_{\mathrm{\infty }}\text{M}$

The pre-atmospheric mass and velocity are represented by m_∞ and v_∞.When the results are expressed in terms of the size of the residual mass following atmospheric ablation as a function of the initial mass and velocity, it is found that the final residual mass is almost independent of the original mass of the meteoroid, but very strongly dependent on the original velocity. For example, the residual mass is very nearly 10^?7 g for a meteoroid with velocity 18 kms^?1 for initial masses from 10^?7 to 10^?3 g. On the other hand, a slight change in the initial velocity to 20 km s^?1 will shift the residual mass to approx. 10^?8 g. This strong velocity dependence coupled with the weak dependence on the original mass has important consequences for the sampling of ablation product micrometeorites.     

Central configurations of four bodies with one inferior mass

The number of equivalence classes of central configurations (abbr. c.c.) in the planar 4-body problem with three arbitrary and a fourth small mass is investigated. These c.c. are derived according to their generic origin in the 3-body problem. It is shown that each 3-body collinear c.c. generates exactly 2 non-collinear c.c. (besides 4 collinear ones) of 4 bodies with smallm ₄0; and that any 3-body equilateral triangle c.c. generates exactly 8 or 9 or 10 (depending onm ₁,m ₂,m ₃) planar 4-body c.c. withm ₄=0. Further, every one of these c.c. can be continued uniquely to sufficiently smallm ₄>0 except when there are just 9; then exactly one of them is degenerate, and we conjecture that it is not continuable tom ₄>0.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

A highly symmetric restricted nine-body problem and the linear stability of its relative equilibria

We study a highly symmetric nine-body problem in which eight positive masses, called the primaries, move four by four, in two concentric circular motions such that their configuration is always a square for each group of four masses. The ninth body being of negligible mass and not influencing the motion of the eight primaries. We assume all the nine masses are in the same plane and that the masses of the primaries are \(m_{1}=m_{2}=m_{3}=m_{4}=\tilde{m}\) and m ₅=m ₆=m ₇=m ₈=m and the radii associated to the circular motion of the bodies with mass \(\tilde{m}\) is λ∈[λ ₀,1] and for the bodies with mass m is 1. We prove the existence of central configurations which characterize such arrangement of the primaries and we study the influence of the parameter λ, the ratio of the radii of the two circles, on the masses m and \(\tilde{m}\) . We use a synodical system of coordinates to eliminate the time dependence on the equations of motion. We show the existence of equilibria solutions symmetrically distributed on the four quadrants and their dependence on the parameter λ. Finally, we show that there can be 13, 17 or 25 equilibria solutions depending on the size of λ and we investigate their linear stability.     

Central Configurations of the Symmetric Restricted 4-Body Problem

We consider the symmetric planar (3 + 1)-body problem with finite masses m ₁ = m ₂ = 1, m ₃ = µ and one small mass m ₄ = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m ₄ > 0.     

Initial mass functions of Wolf-Rayet stars

On the basis of evolutionary tracks on the HR diagram the lower limit of initial mass functions for Wolf-Rayet stars are estimated. The lower limit to the initial masses of the Wolf-Rayet stars seems to be 20M and in this respect there is no significant difference between the WN and WC stars.     

Intermediate-mass black holes in globular clusters

The mass of central bodies in a number of Milky-Way globular clusters is estimated based on the stellar radial-velocity dispersion data. It is assumed that stars located close to the center of the cluster (i.e., to the black hole) rotate about it, have masses on the order of the solar mass, and that the mass of the gravitating center is greater by a factor of 1000. The radial velocities of stars in the vicinity of cluster centers are analyzed for two hypothetical extreme cases: (1) ordered orbital motion of stars about the gravitating center and (2) chaotic orbital motions. The masses inferred for most of the clusters (10²–10⁴ M _⊙) correspond to intermediate-mass black holes. Another important result of this study consists in the determination of the quantity l, the characteristic scale length of the additional spatial dimension. Given the age and mass of the globular cluster NGC 6397 we estimate l to be between 0.02 and 0.14 mm.     

New stacked central configurations for the planar 5-body problem

A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m ₁, m ₂, m ₃ (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m ₄, m ₅ are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.