Potassium diffusion in melilite: Experimental studies and constraints on the thermal history and size of planetesimals hosting CAIs

Abstract— Among the calcium‐aluminum‐rich inclusions (CAIs), excess ⁴¹K (⁴¹K*), which was produced by the decay of the short‐lived radionuclide ⁴¹Ca (t_1/2 = 0.1 Myr), has so far been detected in fassaite and in two grains of melilites. These observations could be used to provide important constraints on the thermal history and size of the planetesimals into which the CAIs were incorporated, provided the diffusion kinetic properties of K in these minerals are known. Thus, we have experimentally determined K diffusion kinetics in the melilite end‐members, åkermanite and gehlenite, as a function of temperature (900–1200 °C) and crystallographic orientation at 1 bar pressure. The closure temperature of K diffusion in melilite, T_c(K:mel), for the observed grain size of melilite in the CAIs and cooling rate of 10–100 °C/Myr, as calculated from our diffusion data, is much higher than that of Mg in anorthite. The latter was calculated from the available Mg diffusion data in anorthite. Assuming that the planetesimals were heated by the decay of ²⁶Al and ⁶⁰Fe, we have calculated the size of a planetesimal as a function of the accretion time t_f such that the peak temperature at a specified radial distance r_c equals T_c(K:mel). The ratio (r_c/R)³ defines the planetesimal volume fraction within which ⁴¹K* in melilite grains would be at least partly disturbed, if these were randomly distributed within a planetesimal. A similar calculation was also carried out to define R versus t_f relation such that ²⁶Mg* was lost from ?50% of randomly distributed anorthite grains, as seems to be suggested by the observational data. These calculations suggest that ?60% of melilite grains should retain ⁴¹K* if ?50% of anorthite grains had retained ²⁶Mg*. Assuming that t_f was not smaller than the time of chondrule formation, our calculations yield minimum planetesimal radius of ?20–30 km, depending on the choice of planetesimal surface temperature and initial abundance of the heat producing isotope ⁶⁰Fe.     

Static electrically charged fluids in terms of pressure: general property

A most general exact solution to the Einstein-Maxwell equations for static charged perfect fluid is sought in terms of pressure. Subsequently, metrics (e ^λ and e ^υ ), matter density and electric intensity are expressible in terms of pressure. Consequently, Pressure is found to be an invertible arbitrary function of ω(=c ₁+c ₂ r ²), where c ₁ and c ₂(≠0) are arbitrary constants, and r is the radius of star, i.e. p=p(ω). We present a general solution for charged pressure fluid in terms for ω. We list and discuss some old and new solutions which fall in this category.     

Charged analogues of Schwarzschild interior solution in terms of pressure

Recently, Bijalwan (Astrophys. Space Sci., doi:, 2011a) discussed charged fluid spheres with pressure while Bijalwan and Gupta (Astrophys. Space Sci. 317, 251–260, 2008) suggested using a monotonically decreasing function f to generate all possible physically viable charged analogues of Schwarzschild interior solutions analytically. They discussed some previously known and new solutions for Schwarzschild parameter u( = \fracGMc²a ) ￡ 0.142u( =\frac{GM}{c^{2}a} ) \le 0.142, a being radius of star. In this paper we investigate wide range of u by generating a class of solutions that are well behaved and suitable for modeling Neutron star charge matter. We have exploited the range u≤0.142 by considering pressure p=p(ω) and f = ( f₀(1 - \fracR²(1 - w)a²) +f_a\fracR²(1 - w)a² )f = ( f_{0}(1 - \frac{R^{2}(1 - \omega )}{a^{2}}) +f_{a}\frac{R^{2}(1 - \omega )}{a^{2}} ), where w = 1 -\fracr²R²\omega = 1 -\frac{r^{2}}{R^{2}} to explore new class of solutions. Hence, class of charged analogues of Schwarzschild interior is found for barotropic equation of state relating the radial pressure to the energy density. The analytical models thus found are well behaved with surface red shift z _s ≤0.181, central red shift z _c ≤0.282, mass to radius ratio M/a≤0.149, total charge to total mass ratio e/M≤0.807 and satisfy Andreasson’s (Commun. Math. Phys. 288, 715–730, 2009) stability condition. Red-shift, velocity of sound and p/c ² ρ are monotonically decreasing towards the surface while adiabatic index is monotonically increasing. The maximum mass found to be 1.512 M _Θ with linear dimension 14.964 km. Class of charged analogues of Schwarzschild interior discussed in this paper doesn’t have neutral counter part. These solutions completely describe interior of a stable Neutron star charge matter since at centre the charge distribution is zero, e/M≤0.807 and a typical neutral Neutron star has mass between 1.35 and about 2.1 solar mass, with a corresponding radius of about 12 km (Kiziltan et al., [astro-ph.GA], 2010).     

Spread of matter over a neutron-star surface during disk accretion: Deceleration of rapid rotation

The accretion of hot slowly rotating gas onto a supermassive black hole is considered. The important case where the velocities of turbulent pulsations at the Bondi radius r _B are low, compared to the speed of sound c _s, is studied. Turbulence is probably responsible for the appearance of random average rotation. Although the angular momentum at r _B is low, it gives rise to the centrifugal barrier at a depth r _c = l ² /GM _BH ≪ r _B, that hinders supersonic accretion. The numerical solution of the problem of hot gas accretion with finite angular momentum is found taking into account electron thermal conductivity and bremsstrahlung energy losses of two temperature plasma for density and temperature near Bondi radius similar to observed in M87 galaxy. The saturation of the Spitzer thermal conductivity was also taken into account. The parameters of the saturated electron thermal conductivity were chosen similar to the parameters used in the numerical simulations of interaction of the strong laser beam radiation with plasma targets. These parameters are confirmed in the experiments. It is shown that joint action of electron thermal conductivity and free-free radiation leads to the effective cooling of accreting plasma and formation of the subsonic settling of accreting gas above the zone of a centrifugal barrier. A toroidal condensation and a hollow funnel that separates the torus from the black hole emerge near the barrier. The barrier divides the flow into two regions: (1) the settling zone with slow subKeplerian rotation and (2) the zone with rapid supersonic nearly Keplerian rotation. Existence of the centrifugal barrier leads to significant decrease of the accretion rate Ṁ in comparison with the critical Bondi solution for γ = 5/3 for the same values of density and temperature of the hot gas near Bondi radius. Shear instabilities in the torus and related friction cause the gas to spread slowly along spirals in the equatorial plane in two directions.As a result, outer (r > r _c) and inner (r < r _c) disks are formed. The gas enters the immediate neighborhood of the black hole or the zone of the internal ADAF flow along the accretion disk (r < r _c). Since the angular momentum is conserved, the outer disk removes outward an excess of angular momentum along with part of the matter falling into the torus. It is possible, that such outer Keplerian disk was observed by Hubble Space Telescope around the nucleus of the M87 galaxy in the optical emission lines. We discuss shortly the characteristic times during which the accretion of the gas with developed turbulence should lead to the changes in the orientation of the torus, accretion disk and, possibly, of the jet.     

Equivalence principle (EP) and solar system constraints on $R(1\pm\epsilon\ln({R \over R_{c}}))$ model of gravity

Experiments on the violation of equivalence principle (EP) and solar system give a number of constraints in which any modified gravity model must satisfy them. We study these constraints on a kind of f(R) gravity as f(R) = R(1±eln([(R)/(R_c)]))f(R) = R(1\pm\epsilon\ln({R \over R_{c}})). For this investigation we use of chameleon mechanism and show that a spherically body has thin-shell in this model. So that we obtain an effective coupling of the fifth force which is suppressed through a chameleon mechanism. Also, we obtain γ _PPN =1±1.13×10⁻⁵ which is agreement with experiment results. At last, we show that for R _c ≈ρ _c this model is consistent with EP, thin shell condition and fifth force of chameleon mechanism for ε⋍10⁻¹⁴.     

The distinctive feature of relativistic stellar systems

In this paper we adopt the method of relativistic fluid dynamics to examine the number density distribution of stars around a massive black hole in the core of stellar clusters. We obtain extensive results,n(r) r ^–a, 3/2a9/2, which include, respectively, then(r) r ^–7/4 power law obtained by Bahcall and Wolf and then(r) r ^–9/4 power law by Peebles. Sincen(r) is not an observable quantity for star clusters, we also consider general relativity effects, i.e., the consequence of the bending of light, in calculating the projected density of stars in such a system. As an example we employ a massive black hole 10³ M inlaid in the center of a globular cluster and calculate various projected densities of stars. The results show that cusp construction occurs in all cases unless the central black hole massM=0, and the polytropic index does not affect at all the position of the capture radiusr _a. The obvious differences in the surface density is only embodied in the interior of the capture radius. At the outer regions of the core, the surface density of stars declines rapidly with ar ^–5 power law in all cases. These results can be applied to cases of unequal-mass and non-steady state.     

The structure of dust tails of comets II. The tail and dust content of comet Arend-Roland

The modified distribution function of dust particles, f(γ), which can be determined from tail brightness profiles on the basis of mechanical theory, is discussed with special regard to its reliability and accuracy. Physical significance of f(γ) is also discussed in terms of dust model parameters, and it is shown that f(γ), if treated carefully, will serve as an effective tool in studying cometary dust. Four isophotes of Comet Arend-Roland, 1957 III, in the orange-red light (λλ 0.53–0.68 micron) obtained by Ceplecha (1958), are analysed by the numerical method described in Paper I (KIMURA and LIU 1975) with some improvements and higher approximations. The distribution f(γ) thus obtained shows a bimodal character with peaks at γ = 0.10 and 0.010 with a relative height ratio of 1 to 0.6. Dust emission rate, which is assumed to follow the inverse square law of heliocentric distance, is estimated to give P_dC_sca = (1.3±0.5) × 10⁹ cm²/sec, where P_d is the rate of particle emission at 1 a.u. and the C_sca is a mean effective cross section of particles for light scattering including the phase effect (the scattering angles of present interest range from about 80 to 100 degrees).     

Time variation of ellipticity of globular clusters in the Large Magellanic Cloud

Dynamical evolution of globular clusters in the Large Magellanic Cloud (LMC) is investigated by means of N-body simulations; particular attention is paid to time evolution in the ellipticitical figure of globular clusters. The simulations were started with a binary globular cluster. It merged into a single cluster with ellipticity of about 0.3. The simulations were continued until the cluster became rounder due to the effects of two body relaxation and of tidal field of LMC. It is found that the outward angular momentum transport due to the gravothermal contraction makes the inner region rounder; the ellipticity at about the initial half-mass radius (r _h) decreases with the e-folding time of 20 relaxation times. On the other hand, the outer region becomes rounder due to the stripping of stars by the tidal field; the ellipticity at about 3r _h decreases with the e-folding time of 80 crossing times therein, though the time scale depends on the direction of the tidal field relative to the spin of the cluster. These two effects are comparable at about the half-mass radius. Taking account of such theoretical results we reanalyzed observed data for the ellipticity at about the half-mass radius of LMC clusters. We estimated the relaxation time and crossing time for each of the observed clusters, from which we calculated the effective time of getting round of the cluster. We plotted the observed ellipticity of the clusters against their non-dimensional age — i.e., the age normalized by the effective time. We found that observed ellipticity distribution is consistent with our picture.     

Quadrature solution for the general relativistic motion of a satellite or a planet

The Schwarzschild field of a central massM is used to derive the general relativistic motion of a particle in a bounded orbit aroundM. A quadrature gives the central angle as a quasi-periodic function (f) of an effective true anomalyf. The linear term is an infinite series, whose second term yields the usual rate of advance of pericenter. For an artificial satellite this may be as large as 17 of arc per year. The periodic part is a sine series, with coefficients containing the small parameter 2GM/c ² p, wherep is closely approximated by the classical semi-latus rectum. The radius vectorr is a Kepler-like function off.The essentially new features of the calculation are the appropriate factoring of a certain cubic polynomialF(p/r), the use of the above effective true anomalyf, and the introduction of an effective eccentric anomalyE. The latter serves to reduce the differential equation forf as a function of the timet, obtained by combining the solution for (f) with the relativistic integrals of motion, to a Kepler equation forE.Knowing the constants of the motion, one can then solve successively forE(t), f(t), r(t), and (t). This is best done as a variational calculation, comparing the relativistic orbiter with a classical orbiter having the same initial conditions. The resulting variations agree with those of Lass and Solloway, but the present method is quite different from theirs and results in a simpler algorithm. The results show that the radial and transverse corrections, r andr , arising from the Schwarzschild field, may be of the order of a kilometer for 1000 revolutions of an Earth satellite of orbital eccentricitye ₀0.6.For bounded motion, the cubic polynomialF(p/r) has three positive real zeros, the two smaller ones corresponding to apocenter and pericenter. The third and apparently non-physical one occurs forrSchwarzschild radius. It may thus correspond to the incipient fall of the orbiter into the central body, if the latter is a black hole.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.Research sponsored by NASA Goddard Space Flight Center under Contract No. NAS 5-11909.     

The time-scale for core collapse in spherical star clusters

The collapse time for a cluster of equal-mass stars is usually stated to be either 330 central relaxation times (t_rc) or 12-19 half-mass relaxation times (t_rh). But the first of these times applies only to the late stage of core collapse, and the second only to low-concentration clusters. To clarify how the time depends on the density profile, the Fokker-Planck equation is solved for the evolution of a variety of isotropic cluster models, including King models, models with power-law density cusps of ρ ∼ r^−γ, and models with nuclei. The collapse times for King models vary considerably with the cluster concentration when expressed in units of t_rc or t_rh, but vary much less when expressed in units of t_rc divided by a dimensionless measure of the temperature gradient in the core. Models with cusps have larger temperature gradients and evolve faster than King models, but not all of them collapse: those with 0 < γ < 2 expand because they start with a temperature inversion. Models with nuclei collapse or expand as the nuclei would in isolation if their central relaxation times are short; otherwise their evolution is more complicated. Suggestions are made for how the results can be applied to globular clusters, galaxies, and clusters of dark objects in the centers of galaxies.Scott D. Tremaine     

Inhomogeneities in the spatial distribution of gamma-ray bursts

The distribution of pairwise distances f(l) for different dependences r(z) of the metric distance is used to reveal inhomogeneities in the spatial distribution of 201 long (T ₉₀>2^s) gamma-ray bursts with measured redshifts z. For a fractal set with dimensionality D, this function behaves asymptotically as f(l) ∼ l ^D−1 for small l. Signs of fractal behavior with dimensionality D = 2.2–2.5 show up in all the models considered for the spatial distribution of the gamma-ray bursts. Several spatially distinct groups of gamma-ray bursts are identified. The group with equatorial coordinates ranging from 23^h56^m to 0^h49^m and δ from +19° to +23° with redshifts of 0.81–0.94 is examined separately.     

Filling factors and scale heights of the diffuse ionized gas in the Milky Way

The combination of dispersion measures of pulsars, distances from the model of Cordes & Lazio (2002) and emission measures from the WHAM survey enabled a statistical study of electron densities and filling factors of the diffuse ionized gas (DIG) in the Milky Way. The emission measures were corrected for absorption and contributions from beyond the pulsar distance. For a sample of 157 pulsars at |b | > 5. and 60° < ℓ < 360°, located in mainly interarm regions within about 3 kpc from the Sun, we find that: (1) The average volume filling factor along the line of sight and the mean density in ionized clouds are inversely correlated: ( ) = (0.0184 ± 0.0011) ^{–1.07 ± 0.03} for the ranges 0.03 < < 2 cm^–3 and 0.8 > > 0.01. This relationship is very tight. The inverse correlation of and causes the well‐known constancy of the average electron density along the line of sight. As (z ) increases with distance from the Galactic plane |z |, the average size of the ionized clouds increases with |z |. (2) For |z| < 0.9 kpc the local density in clouds n _c(z ) and local filling factor f (z ) are inversely correlated because the local electron density n _e(z ) = f (z )n _c(z ) is constant. We suggest that f (z ) reaches a maximum value of >0.3 near |z | = 0.9 kpc, whereas n _c(z ) continues to decrease to higher |z |, thus causing the observed flattening in the distribution of dispersion measures perpendicular to the Galactic plane above this height. (3) For |z | < 0.9 kpc the local distributions n _c(z ), f (z ) and (z ) have the same scale height which is in the range 250 < h ≲ 500 pc. (4) The average degree of ionization of the warm atomic gas (z ) increases towards higher |z | similarly to (z ). Towards |z | = 1 kpc, (z ) = 0.24 ± 0.05 and (z ) = 0.24 ± 0.02. Near |z | = 1 kpc most of the warm, atomic hydrogen is ionized. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Photometry of the poorly studied galactic open star clusters King 13, King 18, King 19, King 20, NGC 136, and NGC 7245

We present the results of our BV R _c I _c CCD photometry for six Galactic open star clusters toward the Perseus spiral armperformed at the Special Astrophysical Observatory of the Russian Academy of Sciences. Based on these data and using JHK _s photometry from the 2MASS catalog, we have determined the ages, distances, and color excesses for the clusters: 710 Myr, 2960₋₃₄₀⁺⁴⁰⁰ pc, 0_· ^m 56 ± 0_· ^m 04 (King 13); 130 Myr, 3010₋₂₈₀⁺³⁰⁰ pc, 0_· ^m 69 ± 0_· ^m 04 (King 18); 560 Myr, 2630₋₂₇₀⁺³¹⁰ pc, 0_· ^m 69 ± 0_· ^m 08 (King 19); 160 Myr, 1750₋₇₀⁺⁸⁰ pc, 0_· ^m 77 ± 0_· ^m 05 (King 20); 250 Myr, 5220₋₃₂₀⁺³⁵⁰ pc, 0_· ^m 70 ± 0_· ^m 09 (NGC 136); 320 Myr, 3390₋₂₀₀⁺²¹⁰ pc, 0_· ^m 43 ± 0_· ^m 03 (NGC 7245).     

Correlation function of quasars from SDSS DR3

We calculate the parameters of the two-point correlation function of quasars w(r) = (r _c /r) ^γ on the basis of the SDSS DR3 data. The correlation functions are first determined from projected distances with the use of a special technique for compiling randomized catalogs. Next the parameters of the spatial correlation function are obtained with the assumption of local isotropy. For the quasars with redshifts z = 0.8–2.1, we obtained the estimates γ = 1.76 ± 0.14, r _c = 6.60 ± 0.85 h ^?1 Mpc in the comoving distance range 2–30 Mpc and γ = 1.90 ± 0.11, r _c = 6.95±0.57 h ^?1 Mpc in the range 2–50 Mpc. These estimates agree, within the limits of errors, with the estimates obtained for the redshifts 0.4 < z < 2.1. The original catalog shows some deficit of pairs with separations less than 1 Mpc.     

The formation of a ring galaxy during a head-on collision between a disk and a spherical galaxy

The tidal force effects of a spherical galaxy passing head-on through a disk galaxy have been studied at various regions of the disk galaxy and for various orientations of the disk galaxy with respect to the direction of relative motion of the two galaxies. The density distribution of the disk galaxy is taken to be, (r)=_ce^–4r/R , where _c is the central density andR is the radius of the disk. The density distribution of the spherical galaxy is taken to be that of a oolytrope of indexn=4. It is found that as a result of the collision, through the central parts and the outer parts of the disk galaxy remain intact, the region in between these two regions disrupts. Thus a ring galaxy with a nucleus embedded in the ring-i.e., a ring galaxy of the RN-type, is formed.     

Ordinary Chondrites: 2. Diffusion Losses of Rare Gases

Data on the isotopic abundances and ratios of light rare gases (He and Ne) in 600 ordinary chondrites are analyzed. The ratio of cosmogenic isotopes (³He/²¹Ne) _c in 20% of the ordinary chondrites has been found to lie well below the correlation line that represents the dependence of (³He/²¹Ne) _c on (²²Ne/²¹Ne) _c . This effect shows up most clearly in ⁴He _r -chondrites, particularly in meteorites with diffusion losses of radiogenic ²¹Ne _c , and is most likely attributable to the predominant (compared to ³He _c ) diffusion losses of cosmogenic ³He_cthrough the solar heating of meteorites in orbits with small perihelion distances. This effect is enhanced by periodic variations in orbital parameters (including the perihelion distance) of meteorites throughout their exposure histories. Thermoluminescence data for ordinary chondrites confirm this scenario. The (³He/²¹Ne) _c ratio for 15% of the chondrites was significantly overestimated, which may stem from the fact that such meteorites were heavily shielded in preatmospheric bodies.     

Sulfuric Acid Production on Europa: The Radiolysis of Sulfur in Water Ice

Europa's surface is chemically altered by radiolysis from energetic charged particle bombardment. It has been suggested that hydrated sulfuric acid (H₂SO₄·nH₂O) is a major surface species and is part of a radiolytic sulfur cycle, where a dynamic equilibrium exists between continuous production and destruction of sulfur polymers S_x, sulfur dioxide SO₂, hydrogen sulfide H₂S, and H₂SO₄·nH₂O. We measured the rate of sulfate anion production for cyclo-octal sulfur grains in frozen water at temperatures, energies, and dose rates appropriate for Europa using energetic electrons. The measured rate is G_Mixture(SO₄²⁻)=f_Sulfur (r₀/r)^βG₁ molecules (100 eV)⁻¹, where f_Sulfur is the sulfur weight fraction, r is the grain radius, r₀=50 μm, β≈1.9, and G₁=0.4±0.1. Equilibrium column densities N are derived for Europa's surface and follow the ordering N(H₂SO₄) » N(S)>N(SO₂)>N(H₂S). The lifetime of a sulfur atom on Europa's surface for radiolysis to H₂SO₄ is τ(−S)=120(r/r₀)^β years. Rapid radiolytic processing hides the identity of the original source of the sulfurous material, but Iogenic plasma ion implantation and an acidic or salty ocean are candidate sources. Sulfate salts, if present, would be decomposed in <3800 years and be rapidly assimilated into the sulfur cycle.     

Radiative cooling flows of self-gravitating filamentary clouds

We study the dynamics of a self-gravitating cooling filamentary cloud using a simplified model. We concentrate on the radial distribution and restrict ourselves to quasi-hydrostatic, cylindrically symmetric cooling flows. For a power-law dependence of cooling function on the temperature, self-similar solutions which describe quasi-hydrostatic cooling flows are derived. We consider obtically thin filaments with a constant mass per unit length and the solutions are parameterized by their line masses. There is no polytropic relation between the density and the pressure. The filament experiences radiative condensation, irrespective of the γ,the gas specific heat ratio. So, the filament becomes denser due to the quasi-hydrostatic flows and the density at the center (ρ_c) increases in proportion to (t ₀-t)^-1, where t denotes the time. The term,t ₀, denotes an epoch at which the central density increases infinitely. We also found that the radius of the filament (r _c) decreases in proportion to (t ₀-t)^1/2. This revised version was published online in July 2006 with corrections to the Cover Date.     

Correlation properties of galaxies from the Main Galaxy Sample of the SDSS survey

The apparatus of correlation gamma function (Γ^*(r)) is used to analyze volume-limited samples from the DR4 Main Galaxy Sample of the SDSS survey with the aim of determining the characteristic scales of galaxy clustering. Up to 20h ^?1 Mpc (H ₀ = 65 km s^?1 Mpc^?1), the distribution of galaxies is described by a power-law density—distance dependence, Γ^*(r) ∝ r ^?γ , with an index γ ≈ 1.0. A change in the state of clustering (a significant deviation from the power law) was found on a scale of (20–25) h ^?1 Mpc. The distribution of SDSS galaxies becomes homogeneous (γ ～ 0) from a scale of ～60h ^?1 Mpc. The dependence of γ on the luminosity of galaxies in volume-limited samples was obtained. The power-law index γ increases with decreasing absolute magnitude of sample galaxies M _abs. At M _abs ～ ?21.4, which corresponds to the characteristic value M _r ^* of the SDSS luminosity function, this dependence exhibits a break followed by a more rapid increase in γ.     

Das Korrespondenzprinzip in der Kosmologie. II. Klassische und relativistische Darstellungen der Weltmodelle

According to the equivalence between the FRIEDMANN equation of relativistic cosmology and the condition for the time-independence H = o of the HAMILTON ian H of an isotropic particle-system in the NEWTON ian mechanics (which equivalence is proved in the part I of our paper) we construct the corresponding classical HAMILTON ians to the relativistic world-models. Each cosmological model which is resulting from a physically meaningful gravitation theory must give a FRIEDMANN equation as the cosmological formulation of the time-independence condition of the energy H for the corresponding NEWTON ian N-particle system. In general relativity, EINSTEIN's field equations are including EINSTEIN's strong principle of equivalence and are giving the constance f = o and M = o of the gravitation-number f and of the mass M of the universe additional to FRIEDMANN's equation. – In special relativity, we have fM = o and this MILNE -universe is possessing a NEWTON ian and a general relativistic interpretation, too. – However, if the postulate together with the “cosmological principle” other principles about the world structure, too (p. e. MACH'S or DIRAC'S principle or the “perfect cosmological principle” by the steady-state cosmology), then EINSTEIN'S weak principle of equivalence can be fulfilled, only. In these world models the gravity-mass fM becomes a function of the cosmic time t [d/dt(fM) ± o] and this variability of fM is compatible with the constance H = o of the energy H of the NEWTON ian particle-system. For flat three-dimensional cosmological spaces (with H = Ḣ = o) a creation of rest-mass (M > o) is possible. This creation is the pecularity of the steady-state cosmos (with M > o, f = o) and of JORDAN'S cosmos (with M > o, f < o). The MACH -EINSTEIN -doctrine about the perfect determination of the inertia and of the space-time-metric by the cosmic gravitation is founded on the substitution of the NEWTON ian HAMILTON ian by a GAUSS -RIEMANN ian gravitation potential U^*(r_AB' v_AB) (TREDER 1972). Therefore, the FRIEDMANN equation for a universe with MACH'S principle is resulting from the analytical expression of the time-independence of this RIEMANNian potential U* = 0. In the case of such MACH-EINSTEIN's-Universes EINSTEIN'S condition 3fM = c8r between the mass A4 and the radius Y of the universe is valid additional to FRIEDMANN'S equation. For these universes, the EINSTEIN condition determinates the instantaneous value of the gravitation-number f. - The explicite form of the conditions H = o or h' = o gives the equation of motion for the cosmic fundamental particles with attraction and repulsion forces, generally.