Models of galactic mass distribution     

K. A. Innanen 《Astrophysics and Space Science》1973,22(2):393-411

Three groups of galactic mass models, each consisting of nine inhomogeneous spheroids of two kinds are described, according to three adopted values of the total density near the Sun: 0.10, 0.15 and 0.20 M pc^–3. Approximately 20% of the total mass of each model is in the halo, constructed to adequately fit recent RR Lyrae star observations. It is shown that the maxima found in the RR Lyrae star densities towards the galactic axis (Plaut, 1970) should not be interpreted as being associated with the galactic nucleus, but as the result of the greater decrease in density with increasingz over the increase in density as the galactic axis is approached. Even at the low galactic latitude of 5° (l=0°), this effect causes a 0.5 kpc correction to the distance to the galactic centre. A basic model for kpc, km s^–1, M pc^–3 is first constructed, mainly to satisfy structural conditions near the sun and in the halo. An attempt to optimize the basic model is made by scaling it so as to retain constant density and angular velocity near the sun, and to best fit kinematic data, including the recent re-examination of the 21-cm data of Simonson and Mader (1972). No unknown matter is required in the models, in accordance with the results of Weistrop (1972b), and, as pointed out earlier (Innanen, 1966b) the faintM-stars must be in a highly flattened spheroid. The optimizing indicates that an adequate fit to kinematics can be achieved for km s^–1. More detailed results are tabulated for a representative model for which . Two new galactic density functions are discussed in the Appendix.  相似文献   

On the role of plasma effects in the cosmic ray propagation and isotropization in the Galaxy     

V. L. Ginzburg  V. S. Ptuskin  V. N. Tsytovich 《Astrophysics and Space Science》1973,21(1):13-38

Cosmic ray (c. r.) propagation in interstellar magnetic fields is often considered in the diffusion approximation, i.e. by the diffusion equation in the coordinate space. Cosmic ray momentum distribution in this case is considered isotropic when the space gradients of c.r density are absent. This approach, with the use of an unfixed effective diffusion coefficientD independent of the energyE enables one to describe all the data available However, neither the diffusion mechanism nor the limits of applicability of the diffusion approximation is clear particularly ifD is independent ofE. Furthermore, the diffusion coefficientD must be expressed through the characteristics of the interstellar medium and possibly through the flux velocity and density of c.r. etc. One of the possible approaches for the analysis of the mechanism and characteristic features of c.r. distribution and isotropization is the account taken of the plasma effects and specifically, the study of c.r. flux instability arising when c.r. are moving in the interstellar plasma. As a result of such instability c.r. may generate waves of different types (magnetohydrodynamic, high-frequency plasma and other waves). Generation of waves and scattering on them result in isotropization of cosmic rays while their propagation under certain conditions turns out similar to that under diffusion.An attempt is made here to systematically analyse the avove mentioned plasma effects and to find out to what extent they are responsible for the behaviour of c.r. in the Galaxy. It turns out that c.r. In any case this is true if this mechanism is regarded as the only c.r. isotropization mechanizm within a wide energy range from 1 to 1000 GeV. Isotropization and spatial diffusion of c.r. up toE100–1000 GeV on the waves from external sources (for example, on the waves from the supernova shells) also proved impossible if the diffusion coefficient is assumed to be independent of c.r. energy. Some new possibilities of c.r. isotropization are also considered.A List of Notations D cosmic ray (c.r.) space diffusion coefficient - degree of c.r. anyisotropy - E,E _kin total and kinetic particle energy - p,p particle momentum and its absolute value - angle between the particle momentum direction and the magnetic field direction (z-axis) - cos - v, particle velocity and its absolute value - c light velocity - f(p),f(E) momentum and energy particle distribution function - N( > E) = N( > p) = f(p) dp/(2)³ = _E f dE c.r. particle density - c.r. spectrum index,N(>E)=KE ^–+1 - n _H neutral particle density - n=n _e=n _i ion and electron density - H niagnetic field - T temperature - thermal velocities of electrons and ions - Boltzmann constant - Alfén velocity - M, m proton and electron masses - e electron charge - wave frequency - _H =eH/Mc, = _H (Mc ²/E) gyrofrequency of a plasma proton and relativistic particle - _H =eH/mc gyrofrequency of an electron - plasma frequency - v _ii,v _ei,v _en,v _in collision frequencies between ions, electrons and ions, electrons and neutrals, ions and neutrals - growth rate of wave amplitude - k,k wave vector and its absolute value - angle between the directions of the vectorsk andH - wave energy density  相似文献   

Sur un modele explicitant les differents types morphologiques des galaxies     

R. Louise 《Astrophysics and Space Science》1982,81(1-2):387-395

In the now classical Lindblad-Lin density-wave theory, the linearization of the collisionless Boltzmann equation is made by assuming the potential functionU expressed in the formU=U ₀ + \(\tilde U\) +... WhereU ₀ is the background axisymmetric potential and \(\tilde U<< U_0 \) . Then the corresponding density distribution is \(\rho = \rho _0 + \tilde \rho (\tilde \rho<< \rho _0 )\) and the linearized equation connecting \(\tilde U\) and the component \(\tilde f\) of the distribution function is given by $$\frac{{\partial \tilde f}}{{\partial t}} + \upsilon \frac{{\partial \tilde f}}{{\partial x}} - \frac{{\partial U_0 }}{{\partial x}} \cdot \frac{{\partial \tilde f}}{{\partial \upsilon }} = \frac{{\partial \tilde U}}{{\partial x}}\frac{{\partial f_0 }}{{\partial \upsilon }}.$$ One looks for spiral self-consistent solutions which also satisfy Poisson's equation $$\nabla ^2 \tilde U = 4\pi G\tilde \rho = 4\pi G\int {\tilde f d\upsilon .} $$ Lin and Shu (1964) have shown that such solutions exist in special cases. In the present work, we adopt anopposite proceeding. Poisson's equation contains two unknown quantities \(\tilde U\) and \(\tilde \rho \) . It could be completelysolved if a second independent equation connecting \(\tilde U\) and \(\tilde \rho \) was known. Such an equation is hopelesslyobtained by direct observational means; the only way is to postulate it in a mathematical form. In a previouswork, Louise (1981) has shown that Poisson's equation accounted for distances of planets in the solar system(following to the Titius-Bode's law revised by Balsano and Hughes (1979)) if the following relation wasassumed $$\rho ^2 = k\frac{{\tilde U}}{{r^2 }} (k = cte).$$ We now postulate again this relation in order to solve Poisson's equation. Then, $$\nabla ^2 \tilde U - \frac{{\alpha ^2 }}{{r^2 }}\tilde U = 0, (\alpha ^2 = 4\pi Gk).$$ The solution is found in a classical way to be of the form $$\tilde U = cte J_v (pr)e^{ - pz} e^{jn\theta } $$ wheren = integer,p =cte andJ _v (pr) = Bessel function with indexv (v ² =n ² + α²). By use of the Hankel function instead ofJ _v (pr) for large values ofr, the spiral structure is found to be given by $$\tilde U = cte e^{ - pz} e^{j[\Phi _v (r) + n\theta ]} , \Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).$$ For small values ofr, \(\tilde U\) = 0: the center of a galaxy is not affected by the density wave which is onlyresponsible of the spiral structure. For various values ofp,n andv, other forms of galaxies can be taken into account: Ring, barred and spiral-barred shapes etc. In order to generalize previous calculations, we further postulateρ ₀ =kU ₀/r ², leading to Poisson'sequation which accounts for the disc population $$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$ AsU ₀ is assumed axisymmetrical, the obvious solution is of the form $$U_0 = \frac{{cte}}{{r^v }}e^{ - pz} , \rho _0 = \frac{{cte}}{{r^{2 + v} }}e^{ - pz} .$$ Finally, Poisson's equation is completely solvable under the assumptionρ =k(U/r ². The general solution,valid for both disc and spiral arm populations, becomes $$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$ The density distribution along the O _z axis is supported by Burstein's (1979) observations.  相似文献   

Periodic solutions for resonance 4/3: Application to the construction of an intermediary orbit for Hyperion’s motion     

I. Stellmacher 《Celestial Mechanics and Dynamical Astronomy》1999,75(3):185-200

The motion of Hyperion is an almost perfect application of second kind and second genius orbit, according to Poincaré’s classification. In order to construct such an orbit, we suppose that Titan’s motion is an elliptical one and that the observed frequencies are such that 4n _H−3n _T+3n _ω=0, where n _H, n _T are the mean motions of Hyperion and Titan, n _ω is the rate of rotation of Hyperion’s pericenter. We admit that the observed motion of Hyperion is a periodic motion such as . Then, .N _H, N _T, k∈ N ⁺. With that hypothesis we show that Hyperion’s orbit tends to a particular periodic solution among the periodic solutions of the Keplerian problem, when Titan’s mass tends to zero. The condition of periodicity allows us to construct this orbit which represents the real motion with a very good approximation. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

Intrinsic orbital coordinates on a Hamiltonian surface     

Michael E. Hough 《Celestial Mechanics and Dynamical Astronomy》1989,47(1):57-85

A reversible dynamical system with two degrees-of-freedom is reduced to a second-order, Hamiltonian system under a change of independent variable. In certain circumstances, the reduced order system may be integrated following an orthogonal curvilinear transformation from Cartesian x,y to intrinsic orbital coordinates , . Solutions for the orbit position and true time variables are expressed by: % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 % da9iaadAgacaGGOaGaeqOVdGNaaiilaiabeE7aOjaacMcacaGGSaGa % aeiiaiaadMhacqGH9aqpcaWGNbGaaiikaiabe67a4jaacYcacqaH3o % aAcaGGPaGaaiilaiaabccacaWGKbGaamiDaiabg2da9iabgglaXoaa % dmaabaWaaSaaaeaacaWGibWaa0baaKqaahaacqaH+oaEaeaacaqGYa % aaaOGaam4raiabgUcaRiaadIeadaqhaaqcbaCaaiabeE7aObqaaiaa % ikdaaaGccaWGfbaabaGaaGOmaiaacIcacaWGibGaey4kaSIaamyvai % aacMcaaaaacaGLBbGaayzxaaWaaWbaaSqabKqaGhaacaaIXaGaai4l % aiaaikdaaaGccaWGKbGaeqiXdqhaaa!6498! \[ x = f(\xi ,\eta ),{\rm{ }}y = g(\xi ,\eta ),{\rm{ }}dt = \pm \left[ {\frac{{_\xi ^{\rm{2}} {\ie} + _\eta ^2 }}{{2( + U)}}} \righ \]1446 1040 where U is the potential function, and z is the new independent variable. The functions f, g may be expressed by quadratures when the metric coefficients {\er},{\ie} are specified. Two second-order, partial differential equations specify {\er}, {\ie} and Hamiltonian {\tH}. Auxiliary conditions are needed because the solutions are underdetermined. For example, both sets of curvilinear coordinate lines are orbits when certain dynamical compatibility conditions between U and {\ie} (or {\er}) are satisfied. Alternatively, when orbits cross the parametric curves, the auxiliary condition {\er} = {\ie} specifies a conformal transformation, and the partial differential equation for {\tH} may be reduced to an ordinary differential equation for the orbit curve. In either case, integrability is guaranteed for Lionville dynamical systems. Specific applications are presented to illustrate direct solution for the orbit (e.g., two fixed centers) and inverse solution for the potential.  相似文献   

Nonlinear newtonian theory and the effect of time dilatation     

Demetrios D. Dionysiou 《Astrophysics and Space Science》1984,104(2):389-397

The fact that the energy density ρ_g of a static spherically symmetric gravitational field acts as a source of gravity, gives us a harmonic function \(f\left( \varphi \right) = e^{\varphi /c^2 } \) , which is determined by the nonlinear differential equation $$\nabla ^2 \varphi = 4\pi k\rho _g = - \frac{1}{{c^2 }}\left( {\nabla \varphi } \right)^2 $$ Furthermore, we formulate the infinitesimal time-interval between a couple of events measured by two different inertial observers, one in a position with potential φ-i.e., dt _φ and the other in a position with potential φ=0-i.e., dt ₀, as $${\text{d}}t_\varphi = f{\text{d}}t_0 .$$ When the principle of equivalence is satisfied, we obtain the well-known effect of time dilatation.  相似文献   

On numerical evaluation of theH-functions of transport problems by kernel approximation for the albedo 0<ω≤1     

Z. Islam  S. R. Das Gupta 《Astrophysics and Space Science》1984,106(2):355-369

Das Gupta represented theH-functions of transport problems for the albedo [0, 1] in the formH(z)=R(z)–S(z) (see Das Gupta, 1977) whereR(z) is a rational function ofz andS(z) is regular on [–1, 0] ^c . In this paper we have representedS(z) through a Fredholm integral equation of the second kind with a symmetric real kernelL(y, z) as . The problem is then solved as an eigenvalue problem. The kernel is converted into a degenerate kernel through finite Taylor's expansion and the integral equation forS(z) takes the form: (which is solved by the usual procedure) where _r 's are the discrete eigenvalues andF _r 's the corresponding eigenfunctions of the real symmetric kernelL(y, z).  相似文献   

Possible semi-annual variation of the Newtonian constant of gravitation     

Asger G. Gasanalizade 《Astrophysics and Space Science》1992,195(2):463-466

A possible semi-annual variation of the Newtonian constant of gravitationG is established. For the aphelion and perihelion points of the Earth's orbit we find, respectively,

相似文献    20. On pulsar-driven isothermal blast-wave models of supernova remanants      I. Lerche 《Astrophysics and Space Science》1977,48(2):335-356 We investigate the ‘equilibrium’ and stability of spherically-symmetric self-similar isothermal blast waves with a continuous post-shock flow velocity expanding into medium whose density varies asr ?ω ahead of the blast wave, and which are powered by a central source (a pulsar) whose power output varies with time ast ω?3. We show that: for ω<0, no physically acceptable self-similar solution exists; for ω>3, no solution exists since the mass swept up by the blast wave is infinite; ? must exceed zero in order that the blast wave expand with time, but ?<2 in order that the central source injects a finite total energy into the blast wave; for 3>ωmin(?)>ω>ωmax(?)>0, where $$\begin{gathered} \omega _{\min } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} + {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} + {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \omega _{\max } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} - {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} - {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \end{gathered} $$ two critical points exist in the flow velocity versus position plane. The physically acceptable solution must pass through the origin with zero flow speed and through the blast wave. It must also pass throughboth critical points if \(\varphi > \tfrac{5}{3}\) , while if \(\varphi< \tfrac{5}{3}\) it must by-pass both critical points. It is shown that such a solution exists but a proper connection at the lower critical point (for ?>5/3) (through whichall solutions pass with thesame slope) has not been established; for 3>ω>ωmin(?) it is shown that the two critical points of (iv) disappear. However a new pair of critical points form. The physically acceptable solution passing with zero flow velocity through the origin and also passing through the blast wave mustby-pass both of the new critical points. It is shown that the solution does indeed do so; for 3>ωmin(?)>ωmax(?)>ω it is shown that the dependence of the self-similar solution on either ω or ? is non-analytic and therefore, inferences drawn from any solutions obtained in ω>ωmax(?) (where the dependence of the solutionis analytic on ω and ?) are not valid when carried over into the domain 3>ωmin(?)>ωmax(?)>ω; all of the physically acceptable self-similar solutions obtained in 3>ω>0 are unstable to short wavelength, small amplitude but nonself-similar radial velocity perturbations near the origin, with a growth which is a power law in time; the physical self-similar solutions are globally unstable in a fully nonlinear sense to radial time-dependent flow patterns. In the limit of long times, the nonlinear growth is a power law in time for 5<ω+2?, logarithmic in time for 5>ω+2?, and the square of the logarithm in time for 5=ω+2?. The results of (vii) and (viii) imply that the memory of the system to initial and boundary values does not decay as time progresses and so the system does not tend to a self-similar form. These results strongly suggest that the evolution of supernova remnants is not according to the self-similar form.  相似文献