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1.
Analytic structure of high-density steady isothermal spheres is discussed using the TOV equation of hydrostatic equilibrium which satisfies an equation of state of the kind:P = K g , = g c 2.Approximate analytical solutions to the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium in (, ), (,U) and (u, v) phase planes in concise and simple form useful for short computer programmes or on small calculator, have been given. In Figures 1, 2, and 3, respectively, we display the qualitative behaviours of the ratio of gas density g to the central density gc , g / gc ; pressureP to the gc ,P/ gc ; and the metric componente –, for three representative general relativistic (GR) isothermal configurations =0.1, 0.2, and 0.3. Figure 4 shows the solution curve (, ) for =0.1, 0.2, and 0.3 (=0 represents the classical (Newtonian) curve). Numerical values of physical quantitiesv (=4r 2 P *(r)), in steps ofu (=M(r)/r)=0.03, and the mass functionU, in steps of =0.2 (dimensionless radial distance), are given, respectively, in Tables I and II. Other interesting features of the configurations, such as ratio of gravitational radius 2GM/c 2 to the coordinate radiusR, mass distributionM(r)/M, pressure (or density) distributionP/P c , binding energy (B.E.), etc., have also been incorporated in the text. It has further been shown that velocity of sound inside the configurations is always less than the velocity of light.Part of the work done at Azerbaijan State University, Baku, U.S.S.R., and Mosul University, Mosul, Iraq, 1985-1986  相似文献   

2.
Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter 2 which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree 12 with respect to which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in 2, ( 2)2 and the coefficient of the term in ( 2)3 which depends upon the initial valuey(0) of the Hill's differential equation.  相似文献   

3.
Savopulos  M.  Quenby  J. J. 《Solar physics》1998,180(1-2):479-485
A numerical solution to the Fisk and Lee (1980) equations for the particle intensity upstream of a corotating interplanetary shock is considered for the November 1991 event observed at Ulysses. A numerically derived parallel diffusion coefficient is available for this region (Quenby et al., 1993), based upon in-situ magnetometer data. Fitting the transport equations solution to the upstream energetic particle distribution function, employing a radial diffusion coefficient = 0 r, where r and are, respectively, radial distance from the Sun and particle velocity, and with 0 fixed from the magnetometer derived coefficient yielded a range of statistically acceptable values of (, ). These ran from (0.5, 0.0) to (1.8, 1.6) along a thin strip of — space, hence demonstrating the improbability that the velocity and radial dependence of the particle diffusion can be fixed from such particle and magnetic field data alone.  相似文献   

4.
Very Large Array (VLA) observations of compact transient sources on the Sun at 2 cm wavelength are presented. These sources have angular sizes of 5–25, brightness temperatures of T B 1–3 × 105 K, and lifetimes ranging between a few minutes to several hours. The emission originates in regions of diffuse plage and quiet Sun, where the photospheric magnetic fields are relatively weak (H 100 G). In some cases the 2 cm radiation may be explained as the thermal bremsstrahlung of a dense (N e 1010 cm-3) plasma in the transition region. For other sources, the relatively high circular polarization ( c 40–50 %) suggests a nonthermal emission mechanism, such as the gyrosynchrotron radiation of mildly relativistic electron with a power-law spectrum.  相似文献   

5.
The purpose of this paper is to study, for small values of , the three-dimensional p–q resonant orbits that are close to periodic second species solutions (SSS) of the restricted three-body problem. The work is based on an analytic study of the in- and out-maps. These maps are associated to follow, under the flow of the problem, initial conditions on a sphere of radius around the small primary, and consider the images of those initial points on the same sphere. The out-map is associated to follow the flow forward in time and the in-map backwards. For both mappings we give analytical expressions in powers of the mass parameter. Once these expressions are obtained, we proceed to the study of the matching equations between both, obtaining initial conditions of orbits that will be 'periodic' with an error of the order 1–, for some (1/3,1/2). Since, as 0, the inner solution and the outer solution will collide with the small primary, these orbits will be close to SSS.  相似文献   

6.
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.  相似文献   

7.
A model of a first generation intermediate star of 5M , with Z=0 has been considered. The model is at an advanced stage of its evolution and has a double shell burning. It burns helium in the inner shell, and hydrogen, via CNO cycle, in the outer shell. =(log/log) T and T =(log/logT) were computed allowing for the oscillations of the relative mass abundance of the reagents in nuclear reactions. Including =(log/log) T and =(log/logT) of mean molecular weight and the effect of the oscillations of abundances due to nuclear reactions, stability was studied. Contrary to the results of the static calculations, we found that instability due to the excitation mechanism provided by the high temperature sensitivity of energy generation rate propagates up to the surface. Thus the model in question was found to be unstable against radial adiabatic pulsations, in its fundamental mode.  相似文献   

8.
The area preserving mapping x = x + a(y – y 3), y = y – a(x – x3), for 0.3 a 2.0 has been studied to locate approximately the x-axis points bounding almost stable regions. For each value of a, these are fixed points with variational trace just greater than 2.0. Transition to chaos can occur rapidly as a increases (with n/k fixed).  相似文献   

9.
A semi-continuous hierarchy, (i.e., one in which there are galaxies outside clusters, clusters outside superclusters etc.), is examined using an expression of the field equations of general relativity in a form due to Podurets, Misner and Sharp. It is shown (a) that for a sufficiently populous hierarchy, the thinning factor( i+1/ i [r i /r i+1] is approximately equal to the exponentN in a continuous density law (=aR –N) provided (r i /r i+1)3-1; (b) that a hierarchical Universe will not look decidedly asymmetric to an observer like a human being because such salient observers live close to the densest elements of the hierarchy (viz stars), the probability of the Universe looking spherically symmetric (dipole anisotropy0.1 to such an observer being of order unity; (c) the existence of a semi-continuous or continuous hierarchy (Peebles) requires that 2 if galaxies, not presently bound to clusters were once members of such systems; (d) there are now in existence no less than ten arguments for believing 2, though recent number counts by Sandageet al. seem to be in contradiction to such a value; (e) Hubble's law, withH independent of distance, can be proved approximately in a relativistic hierarchy provided (i)N=2, (ii)2GM(R)/c 2 R1; (iii)Rc (iv)M0 in a system of massM, sizeR (f) Hubble's law holds also in a hierarchy with density jumps; (g)H100 km s–1 Mpc–1; (h) objects forming the stellar level of the hierarchy (in a cosmology of the Wilson type) must once have had 2GM/c 2 R1; (i) there is a finite pressurep=2Ga in all astrophysical systems (a=R N ,N2); (j) for the Galaxy, theory predictsp G7×10–12 dyn cm–2, observation givesp G5×10–12 dyn cm–2; (k) if the mass-defect (or excess binding energy) hypothesis is taken as a postulate, all non-collapsed astrophysical systems must be non-static, and any non-static, p0 systems must in any case be losing mass; (1) the predicted mass-loss rate from the Sun is 1012 g s–1, compared to 1011 g s–1 in the observed solar wind; (m) the mass-loss rates known by observation imply timescales of 5×109 years for the Sun and 1010 years for other astrophysical systems; (n) degenerate superdense objects composed of fermions must haveN-2 if they were ever at their Schwarzschild radii and comprised a finite numberN B of baryons; (o)N B1057N for degenerate fermion and boson systems; (p)285-4; (q) the metric coefficients for superdense bodies give equations of motion that imply equal maximum luminosities for all evolving superdense bodies (L max1059 erg s–1); (r) larger bodies have longer time-scales of energy radiation atL max (10–5 s for stars,1 h for QSO's) (s) expansion velocities are c soon after the initial loss of equilibrium in a superdense object; (t) if the density parametera(t) in aR –N isa=a (non-atomic constants of physicsc, G, A), andA, thenN=2; (u) N2 is necessary to giveMM at the stellar level of the hierarchy;(v) systems larger than, and including, galaxies must have formed by clumping of smaller systems and not (as advocated by Wertz and others) in a multiple big bang.  相似文献   

10.
Relativistic, isentropic, homogeneous models are constructed by a method that automatically detects instabilities, and evolutionary tracks of central conditions are shown on a (T, ) diagram. Models heavier than 20M become unstable because of pair creation. Iron photodisintegration causes instability in the mass range between 1.5M and 20M . General relativistic effects bring about the onset of instability in models of 1.2–1.5M when the central density is about 1010 g/cm3. Lighter models become white dwarfs. It is pointed out that general relativistic instability will prevent the formation of neutron stars through hydrostatic evolution and may be relevant in setting off low-mass supernovae.  相似文献   

11.
Hickson's compact galaxy groups were classified using the statistical criterion which includes the radial velocities of galaxies as well as their relative positions. These groups on the whole and their components are identified as the confident and probable non-chance ones as well as probable and confident chance ones. All confident chance objects have the discordant radial velocities (the differences of radial velocitiesDV [1000; 20000] km s–1). The special class of objects bright discordants is selected. These galaxies have the discordant radial velocities withDV [825; 8440] km s–1 and have a strong tendency to be the brightest components of their groups. The lowest difference of radial velocities for the last class of objects DV = (1.0±0.2) × 103 km s–1 and we accept this value of DV as the lowest value of discordant radial velocities. It is found that the biggest part of Hickson's compact groups consist of non-chance aggregations of galaxies and some of the cases of discordant-redshifts require a special study in order to explain their origin from a dynamic or some other point of view.Present address.  相似文献   

12.
An exact solution of Einstein's equation is stated in which the density (), pressure (p), scale factorS and metric coefficients are functions of only one dimensionless self-similar variable,ct/R, wheret is cosmic time andR is a co-moving radial coordinate. The solution represents a cosmology that begins as a static sphere having R –2 and evolves into an expanding model which is pressure-free and has a hierarchical type of density law ( R –, approximately, with =a number, 02). It is suggested that this model should supersede the previous models of Wesson and other workers, since it appears to be the simplest cosmology for a hierarchy.  相似文献   

13.
E. Rolli  A. Magun 《Solar physics》1995,160(1):29-40
The analysis of the dynamic evolution of the chromospheric electron density during solar flares is fundamental for the testing of solar flare models. For this purpose we developed a digital imaging spectrograph for the observation of higher Balmer lines below 400 nm with a time resolution of 1 s and an algorithm for the determination of the electron density from the observed line profiles. On January 5, 1992 a M1/1N flare was observed in H, H and Caii H and the temporal evolution of the electron density was determined. The chromospheric electron density rises several times from less than 3 × 1019 to 1 × 1020 m–3 during the hard X-ray peaks.  相似文献   

14.
The brightness distribution of the quiet Sun at 8.6 mm wavelength is synthesized from off-meridian observations using an eight element east-west interferometer with a maximum base line of 16.38 m (1913). The observed brightness distribution is practically flat from the disk center to the optical limb. The effective radius of the nearly uniform component is 1.01 R . If the limb brightening is present, the brightening located between 0.95 R and 1.01 R , and the total flux density of the limb brightening is less than 1% of the total flux density of the Sun. In addition to the nearly uniform component there exists a coronal component just outside the optical limb.  相似文献   

15.
In the first part of this paper [Marchal, Yoshida, Sun Yi-Sui 1985] we have analyzed three-body systems satisfying the condition rkR where k is a suitable constant, r the mutual distance of the two masses of the binary and R the distance between the center of mass of the binary and the third mass.That condition rkR puts limits on the acceleration of the third mass and these limits allow us to determine the corresponding escape velocities.In this second part we look for initial conditions under which the inequality rkR will remain forever satisfied and we develop the corresponding tests of escape and their applications.This leads to a major improvement of the knowledge of the nature of three-body motions especially in the vicinity of triple close approaches.The region of bounded motions is much smaller than was generally expected and numerical computations of particular solutions show that we approach very near to the true limit.  相似文献   

16.
By solving the Euler hydrodynamical equations we have obtained closed form solutions for the angular velocities and pressures of a three stratified non-confocal heterogeneous oblate spheroid. Limiting and particular solutions cases, such as a spheroid with N layers, a stratified spheroid with the same eccentricities, as well as confocal layered spheroids are also explicitly written down. As an application, we have numerically estimated planet Earth's outer and inner cores' ellipticities to be o=1/413.318 and i=1/424.616, respectively. These Earth's ellipticities values are in good agreement with those found in the literature.  相似文献   

17.
. . ,e, , . . e, . , .
Stability of the librational triangular points of the three-dimensional elliptic restricted three-body problem is studied. The problem is solved in the non-linear statement at the small values of eccentricity.For all values ofe, , besides ones which correspond to the resonances of the third and the fourth order the librational points are stable taking into account the terms up to the fourth order in the normal form of the Hamiltonian function of the perturbed motion.At sufficiently smalle and the non-stability in sense of Liapunov has been proved. The approximate equations of the boundary of the stability area in the planee, has been obtained. The cause of the non-stability is an equality of the rotational period of the principal attracting masses in the elliptic orbit and the period of oscillation of indefinitely small mass along the direction perpendicular to the plane of their motion.
  相似文献   

18.
The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR eff=R/ given by the stellar radii and the coefficients for the inner structure of the stars.As seen in a previous work (Paper I), during the lifetime in the Main SequenceR eff(t) remains a near invariant when compared to the variation in the time ofR(t) and (t).With such an effectiveR eff one obtains the amounts of actionA c(M), the effective densities eff(M)=(M)3(M), the densities of action and of energy (or mean presures in the stellar interior)a c(M),e c(M), and the potential energiesE p(M).The amounts of action areA cM k withk1.87 for the M stars,k5/3 for the KGF stars, andk1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek5/3 means a near invariant effective density eff for the KGF stars, while for such stars the mean densities and coefficients present the strongest variations with masses (M)M –1.81, (M)M0.6.The cases for the M stars (e c(M)M –1) and for the A and earlier stars (betweena c(M)=constant and eff(M)M –1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M M25M .With all this, one can build dynamical HR diagrams withA c(M), Ep(M), eff M –p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs.The comparison of the potential energiesE p(M)M –p according to the stellar models used here and the observed frequency function (MM –q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE p(M)(M) is a constant, but this must be confirmed with further studies of the function (M) and its fine structure.There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.  相似文献   

19.
Hong Wei Li 《Solar physics》1986,104(1):131-136
The variations of the growth rates of ECM at first four harmonics in X-, Z-, and O-modes excited by a hollow beam distribution of weakly relativistic electrons with a parameter p / e are presented in this paper. The results show that the second harmonic of ECM in X-mode dominates the instability if < 1, and if 1.2 , 2 or 2.2 3 the third or fourth harmonic will dominate. The second and third harmonics of Z-mode waves grow faster only if 2 2.2 and 3 3.2, respectively, so it would not be a competition in most cases. It is suggested that the ECM emission at these harmonics in X-mode is a possible mechanism to produce solar spike emissions with high brightness temperature at shorter and longer decimetric wavelengths.Proceedings of the Workshop on Radio Continua during Solar Flares, held at Duino (Trieste), Italy, 27–31 May, 1985.On leave from the Department of Astronomy, Nanjing University, Nanjing, The People's Republic of China.  相似文献   

20.
R. Grant Athay 《Solar physics》1988,116(2):223-237
An attempt is made in this paper to determine the coefficient a in a power-law relationship of the form V ~T between the r.m.s. velocity fluctuation, V for raster images with 3 resolution and the temperature, T of line formation using SMM solar data. For T between 8000 and 105 K, the data suggest a best fit with 3/4 < 1. It is argued, however, that unresolved fine structure tends to reduce the observed value of V and that higher resolution data may yield different values for . Skylab data have shown that the non-thermal line broadening velocity, , is proportional to T 1/2. Also, for all temperatures less than 105 K, V . This latter result, however, is again dependent on spatial resolution and may not be true in observations made with sufficient spatial resolution. The magnitudes of both V and indicate that bulk motions play important roles in the structure of the solar atmosphere as well as in its energy and momentum balance. It is important, therefore, to identify the true nature of such motions with better accuracy than is possible with currently available data.The National Center for Atmospheric Research is sponsored by the National Science Foundation.  相似文献   

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