Proper Mass Research Articles

General relativistic stellar stability

We give here a derivation of the general relativistic non adiabatic radial pulsation equation for spherically symmetric non rotating stars. The influence of general relativistic corrections on the non adiabatic terms is treated up to the first order in the relativistic parameter. This implies that the unperturbed model must be considered in a state of "quasi-equilibrium" in the sense that its physical characteristics are slowly varying functions of time, due to the conversion of proper mass into energy, caused by the thermonuclear reactions. Eckart's formalism is used to take into account the presence of a radiative or convective flux but the influence of the radiative viscosity terms is neglected in the derivation of the structure and pulsation equations. Relativistic expressions are given for the vibrational stability coefficient and the secular root, the last one being used to recover and clarify the general relativistic secular instability of supermassive stars above a certain critical mass, found by Appenzeller and Kippenhahn.

A co-accretional model of satellite formation

Numerical calculation of a simple accretion model including the effects of tidal friction indicate that coformation is tenable only if the planet's Q is less than about 10 3. The parameter which most strongly affects the final mass ratio of the pair is the time at which the secondary embryo is introduced. Our model yields the proper Moon-Earth mass ratio if the Moon embryo is introduced when the Earth is only about 1 10 of its final mass. The lunar orbit remains at about 10 Earth radii throughout most of the growth. This model of satellite formation overcomes two difficulties of the “circumterrestrial cloud” model of Ruskol (1960, 1963, 1972): (1) The difficulty of accumulating a mass as great as the entire Moon before gravitational instability reduces the cloud to a small number of moonlets is removed. (2) The differences between terrestrial and outer planet satellite systems is easily understood in terms of the differences in Q between these planets. The high Q of the outer planets does not allow a satellite embryo to survive a significant portion of the accretion process, thus only small bodies which formed very late in the accumulation of the planet remain as satellites. The low Q of the terrestrial planets allows satellite embryos of these planets to survive during accretion, thus massive satellites such as the Earth's Moon are expected. The present lack of such satellites of the other terrestrial planets may be the result of tidal evolution, either infall following primary despinning (Burns, 1973) or escape due to increase in orbit eccentricity.

Premature field evaporation in the atom probe

The motion of ions evaporating in the subnanosecond period before a pulse has matured is invoked in the interpretation of resolution limiting energy deficits in the TOF atom-probe. The second order, nonlinear differential equation of motion in the time and space dependent field is numerically solved by computer. Two shapes of the pulse front, in which the final voltage level is approached either from below or from an overshoot are considered. The latter case accounts for the actual observations by giving the proper magnitude and mass dependence of the energy deficits.

On ellipsoidal solutions of Liouville's equation in general relativity

A relativistic, collisionless gas of gravitating particles all having the same proper mass (possibly equal to zero) is studied under the assumption that the oneparticle distribution function is locally ellipsoidal in momentum space with respect to some timelike vector field (observer). Liouville's equation implies that the distribution function depends only on a quadratic form in the 4- momenta, whose coefficients are a Killing tensor in the case of non- vanishing proper mass, and a conformal Killing tensor in the case of vanishing rest mass of the particles. It is suggested that cosmological models of Bianchi-type I can be described in terms of ellipsoidal momentum distribution functions whose ellipsoidal tensor is built out of the Killing vectors associated with the spatial homogeneity.

The locomotion of elongated bodies in pipes

The theory describing the swimming mechanism of an elongated body is extended to cover the cases of locomotion through unsteady streams in pipes. Such an extension is essential for the artificial fish ‘Pod’, a medical device which swims in the patient's blood vessel. Two approaches are considered. First, potential theory is considered, and the results achieved show that the main influence of the pipe is on evaluation of the proper virtual mass. Next the flow is assumed to be viscous. The consideration of viscosity is obviously necessary for flows in pipes. In that case the virtual mass is replaced by another equivalent mass depending on the viscosity and on the angular frequency of the lateral motion and in addition new terms appear in the local lift expressions. These are recognized as the viscous damping force.

AUTOROTATION, SELF‐STABILITY, AND STRUCTURE OF SINGLE‐WINGED FRUITS AND SEEDS (SAMARAS) WITH COMPARATIVE REMARKS ON ANIMAL FLIGHT

SUMMARY A samara is a winged fruit or seed that autorotates when falling, thereby reducing the sinking speed of the diaspore and increasing the distance it may be transported by winds. Samaras have evolved independently in a large number of plants. Aerodynamical, mechanical, and structural properties crucial for the inherent self‐stability are analysed, and formulae for calculation of performance data are given. The momentum theorem is applied to samaras to calculate induced air velocities. As a basis for blade element analysis, and for directional stability analysis, various velocity components are put together into resultant relative air velocities normal to the blade's span axis for a samara in vertical autorotation and also in autorotation with side‐slip. When falling, a samara is free to move in any sense, but in autorotation it possesses static and dynamic stability. Mainly qualitative aspects on static stability are pre sented. Simple experiments on flat plates at Reynolds numbers about 2000 as in samaras, showed that pitch stability prevails when the C. M. (centre of mass) is located 27–35 % of the chord behind the leading edge. The aerodynamic c.p. (centre of pressure) moves forward upon a decrease of the angle of attack, backward upon an increase. In samara blades the c.m. lies ca. one‐third chord behind the leading edge, and hence the aerodynamic and centrifugal forces interact so as to give pitch stability, involving stability of the angles of attack and gliding angles. Photographs show that the centre of rotation of the samara approximately coincides with its c.m. The coning angle (blade angle to tip path plane) taken up by the samara is determined by opposing moments set up by the centrifugal and aerodynamic forces. It is essentially the centrifugal moment (being a tangent function of the coning angle, which is small) that changes upon a change of coning angle, until the centrifugal and aerodynamic moments cancel out at the equilibrium coning angle. Directional stability is maintained by keeping the tip path plane horizontal whereby a vertical descent path relative to the ambient air is maintained. Tilting of the tip path plane results in side‐slip. Side‐slip leads to an increased relative air speed at the blade when advancing, a reduced speed when retreating. The correspondingly fluctuating aerodynamic force and the gyroscopic action of the samara lead to restoring moments that bring the tip path plane back to the horizontal. Entrance into autorotation is due to interaction between aerodynamic forces, the force of gravity, and inertial forces (when the blade accelerates towards a trailing position behind the c.m. of the samara). The mass distribution must be such that the c.m. lies 0–30 % of the span from one end. In Acer and Plcea samaras the C.M. lies 10–20% from one end, thereby making the disk area swept by the blade large and the sinking speed low. The blade plan‐form is discussed in relation to aerodynamics. The width is largest far out on the blade where the relative air velocities are large. The large width of the blade contributes to a high Re number and thus probably to a better L/D (lift/drag) ratio and a slower descent. The concentration of vascular bundles at the leading edge of the blade and the tapering of the blade thickness towards the trailing edge are essential for a proper chord wise mass distribution. Data are given for samaras of Acer and Plcea, and calculations of performance are made by means of the formulae given in the paper. Some figures for an Acer samara are: sinking speed 0.9 m/sec, tip path inclination 15°, average total force coefficient 1.7 (which is discussed), and a L/D ratio of the blade approximately 3. The performances of samaras are compared with those of insects, birds, bats, a flat plate, and a parachute. They show the samara to be a relatively very efficient structure in braking the sinking speed of the diaspore. In samaras the mass, aerodynamic, and torsion axes coincide, whereas in insect wings the torsicn axis often lies ahead of the other two. Location of the torsion axis in front of the aerodynamic axis in insects tends towards passive wing twisting and passive adjustment of the angles of attack relative to the incident air stream, the direction of which varies along the wing because of wing flapping. Location of the mass axis behind the torsion axis may lead to unfavourable

Attitude stabilisation of the Geos-C spacecraft

GEOS-C will be the third of a series of spacecraft designed for the National Geodetic Satellite Programme. The two earlier and successful spacecraft were designated GEOS-I and II. In these, geocentric stabilisation was acheived to optimise the usage of the radiated and reflected radio and optical power of the geodetic observations. It was the atmospheric attenuation of the optical beacons, by which the spacecraft could be photographed against a star background, which imposed the accuracy requirement of earth pointing to within 5°. Descriptions as well as performance analyses of these attitude control systems have been published. In each, the proper mass distribution was obtained through an end mass on a de Havilland extendible boom. Since initial misorientation and perturbing torques induce motion about the equilibrium orientation in a gravity-gradient system, a mechanism for energy dissipation is necessary. A Magnetically Anchored Eddy Current Damper was used for this purpose. It consists of a magnet assembly that is free to lock onto the geomagnetic field and rotate freely within a conducting spherical shell. Relative motion of the two produces eddy currents in the conductor which in turn impede the relative motion.

Структура источнико в полей Вейлевского т ипа в общей теории относительно сти

Weyl-type fields are those static electrovac solutions to the Einstein-Maxwell field equations for which the metric component g44 and the electrostatic potential Ψ are related by —g44= 1 — 2cΨ+gY2. In the present paper we investigate the structure of the sources that produce Weyl-type fields. It is found that, if the Weyl relationship holds also inside these sources, then they are structured such that over the extent of the source the ratio of the total mass density to the charge density is equal to the constant c in the Woyl relationship. In our result, the total mass density is that function which, when integrated over the extent of the source, gives the Schwarzschild mass as seen at infinity. It is not equal to the proper mass density — (κ/8π)M44, but is equal to the proper mass density plus the binding energy density.

Non-velocity Redshifts and Photon–Photon Interactions

RECENT data on galactic redshift anomalies have led Hoyle and Narlikar1, for example, to propose a qualitative explanation in terms of spatio-temporal variation of proper mass. The consequences of hypotheses such as this are evidently worthy of study, but there are other redshift anomalies, which should also be considered in this context. A possible alternative explanation is in terms of an inelastic photon–photon interaction; this has the advantages of being more inclusive, more quantitative (at least in order of magnitude) and immediately testable in the laboratory.

Mode shapes and frequencies by finite element method using consistent and lumped masses

The rate of convergence of the mode shapes and frequencies by the finite element method using consistent and lumped mass formulations has been established. Simple examples are given to demonstrate the results. It has been found that for a system of differential equations of second order such as the equations of equilibrium in terms of displacement in the theory of elasticity, membrane etc., a proper lumped mass formulation will not suffer any loss of rate of convergence utilizing simple elements. However, in the case of higher order differential equations or when the use of more complicated elements is required or desired, a consistent mass formulation often will provide a better rate of convergence.

Singularities in general relativity

The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.

Theory of Particles with Variable Mass. I. Formalism

The equivalence principle (through the mechanism of the gravitational red shift) allows one to set up a classical formalism with the proper time as an extra degree of freedom, independent of the coordinate time, and with an immediate physical interpretation. Then proper time and mass occur as conjugate variables in a canonical formalism, leading to a gravitational theory of particles with variable mass. The nonrelativistic theory and a relativistic vector theory of gravity are described as models. The theory is capable of providing a dynamical framework for cosmological models with the creation of matter. Some simple examples are discussed, including the steady-state universe with continuous creation, where the correct relation between the density of matter and the Hubble constant appears automatically, with no free parameters.

Singularities in general relativity

The problem of singularities is examined from the standpoint of a local observer. A singularity is defined as a state with an infinite proper rest mass density. It is proved that any inhomogeneity and anisotropy in the distribution and motion of a nonrotating ideal fluid accelerates collapse. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In order to investigate the influence of rotation on the existence of singularities in incoherent matter the Einstein equations together with their first integrals are written out for the points on a vortex filament. They show that rotation decelerates the contraction of space not only in the direction perpendicular to the vector of the angular velocity, but indirectly also along this vector and can prevent the occurrence of a singularity. This conclusion is confirmed by the numerical integration of the Einstein equations. The paper concludes with a discussion of some cosmological implications.

Does a rod, pushed by a force, accelerate less than the same rod pulled by the same force?

The answer is yes, if we consider the mean accelerations of the rod’s points. Namely, the acceleration of the application point of the external force only depends on the rod’s mass and the force’s magnitude. Consequently the other points of the rod accelerate differently according to their positions in order to set up Lorentz contraction. In Sect.1 the problem is solved kinematically, considering that the rod’s motion is rigid if the external force is constant and assuming that the motion of the application point of the external force is hyperbolic. In Sect.2 the above assumption then proves to be in agreement with the relativistic equations of motion for elastic continua. Moreover, by these equations, the acceleration of the application of the external force turns out to be given, in any instantaneous rest system of inertia, byF/m0, whereF is the force’s magnitude andm0 the rod’s proper mass. In Sect.3 the problem is studied also by the equations recently proposed by Cavalleri and Salgarelli in the context of a new formulation called « asynchronous », to distinguish it from the usual « synchronous » formulation of the relativistic dynamics for extended bodies. By the asynchronous equations, the hyperbolic motion for the rod can be immediately deduced. The problem studied can interest models of extended particles, « pushed or pulled » by exchanges of real and virtual photons, mesons, gravitons.

Precise Mass Calibration Method for Mass Spectrograph II Doublet Scaling Method with Consistent Artificial Mass Scale

When there are many doublets of the same ion species, of which fractional mass difference is ro, it is possible to construct consistent artficial mass scales in terms of ro by only position measurements of these doublet lines. With one of these mass scale, the author studies on two possibilities of the mass calibration method suitable for a very high dispersion type mass spectrograph without any sacrifices for precisions of calibrations, which are, at least, the same order of magnitude for those of a single doublet separations, …the general mass center and the doublet scaling method. The former is rather useful for estimations dispersion constants α1 and α2 which are main contributions to the correction terms in an artificial mass scale. Preliminary experimental studies on the latter method are carried out to a few spectra photographed with the Bainbridge-Jordan type apparatus, though these spectra are rather unsuitable even for the preliminary analyses. The doublet scaling method would be a proper mass calibration method for spectra on a photographic plate.

PHOTONS IN THE GRAVITATIONAL FIELD

In a previous paper, "The Gravitokinetic Field and the Orbit of Mercury," a new theory of gravitation was introduced, according to which space–time is always flat and the gravitational field is described by equations of Maxwellian form. In this paper it is shown that the theory correctly predicts the gravitational red shift and the gravitational deflection of a light ray. The interaction of photons with a gravitational field follows from the basic premises of quantum theory, that photon frequency is proportional to its energy and that photon wavelength is inversely proportional to its momentum. The photon velocity and proper mass depend on the gravitational potential, and the deflection of a light ray is due to gravitational refraction. The validity of the antisymmetric field equations for sources of variable rest mass is due to the divergence of the group velocity from the dynamical velocity.

Energy of Infinitely Long, Cylindrically Symmetric Systems in General Relativity

A definition of is proposed for systems invariant under rotations about, and translations along, a symmetry axis. This (which is called energy or energy) takes the form of a covariant vector Pi, which obeys the conservation law Pi; i=0. C is localizable and locally measurable: The component of Pi along the world line of an observer is the C-energy density he measures. Near the symmetry axis of a static system, where strong gravitational fields are absent, C-energy density reduces to proper mass density T00. C is propagated by Einstein-Rosen gravitational waves and by cylindrical electromagnetic waves. In vacuo and in the presence of electromagnetic fields the C on a space-like hyper-surface is minimized when the system is static; and the difference between the potential part and the kinetic part is a Lagrangian for the Einstein-Maxwell field equations. C can be a powerful tool in the analysis of finite as well as infinite cylindrically symmetric systems. Here it is used to elucidate the nature of Einstein-Rosen gravitational radiation, and to suggest and support the conjecture of flux resistance to gravitational collapse: In any configuration of electromagnetic fields collapsing toward a singularity, each electric and magnetic-field line is either entirely ejected from the collapsing region or entirely swallowed by it as collapse proceeds; there can be no flux threading a collapsed region.

Neutron Star Models.

view Abstract Citations (107) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Neutron Star Models. Cameron, A. G. Abstract Previous models of neutron stars were constructed with the assumption that the equation of state of a neutron gas is that of non4nteracting Fermi gas. Such models have a maximum observable mass of about 0.7 solar mass. In fact, the potential energy of a neutron gas depends on the density; this introduces additional terms into the equation of state. A revised equation of state has been derived which makes use of a mean nuclear potential recently given by T. H. R. Skyrme. Twenty neutron star models have been constructed by integrating the general relativistic equations of hydrostatic equilihrium of the neutron gas. The results show that there is an upper limit to the observahle mass of about 2 solar masses; the corresponding upper limit to the proper mass is about 3 solar masses. There is a lower limit to each of these masses of about 0.05 solar mass, below which the neutron star is unstable against transformation into an iron star. The radii of these neutron stars lie in the range 7-9 km. A qualitative discussion of the effects of transformation of neutrons into hyperons at very high densities is given. Publication: The Astrophysical Journal Pub Date: November 1959 DOI: 10.1086/146780 Bibcode: 1959ApJ...130..884C full text sources ADS |

Relativistic Hydrodynamics of the Dirac Matter

The hydrodynamical model of Dirac matter is formulated and clarified. The basic equations are cast in various forms. Some characteristic features of the hydrodynamics are the distinctions between proper mass density and rest particle density and also between particle momentum and velocity, primarily specified by the theta behaviors. The energy-momentum conservation law also manifests a new structure. This quantum effect is interpreted as mechanical stress and flow of heat taking place inside the fluid. The theory provides a new directly physical point of view concerning various transformation properties of the Dirac field. Furthermore it reveals a conspicious quasisymmetrical property of the Dirac field existing between velocity and spin. The theory is formulated for cases of Dirac matter under external electromagnetic field and also of interacting Dirac and electromagnetic fields. It is manifestly gage-independent in both cases. The theory is worked out only for the case of c-number Dirac field. The mathematical method is systematized to estublish how the Dirac field can be manipulated solely with the set of tensor quantities which are related to the Dirac spinor as its bilinear covariants. (N.W.R.)

Lateral Structure of Extensive Air Showers

The problem presented by air showers which have particle densities \ensuremath{\sim}500 per sq meter around the shower axis and which have no multiple peaks with separations of more than a meter has been analyzed previously by Hazen et al. using Fermi's model of meson production. The object of this paper is to show that, contrary to the conclusion of Hazen et al., Fermi's model cannot generate such showers. On the other hand, we show that a modification by Bhabha, to the phenomenological models of meson production, which predicts greater angular concentration of the mesons produced in nucleon-nucleon collisions, can generate the showers of the type referred to above for a suitable choice of the parameter $\ensuremath{\epsilon}$, which in this theory is the ratio of field mass to proper mass.