Improbability of collinear solutions in the n-body problem with generalized attraction law

It is shown that in the n-body problem with generalized attraction law (inverse (α + l )-power of the distance, α > 0) the set of initial conditions which lead to collinear motion is of Lebesgue measure zero and nowhere dense with respect to the set of initial conditions that define solutions in R³ or R².     

The moon’s permanent magnetic field: a cratered-shell model

As part of our study of the larger-scale remanent magnetic field of the Moon, we have examined the effects of cratering in an otherwise spherically symmetrical shell magnetized by a concentric dipolar magnetic fieldH _o to an intensity of magnetizationc H _o, wherec is a constant. In our initial model, we assume that the material excavated from the craters is distributed with random orientation and thus does not contribute to the remanent dipole momentM _g . We further assume that the mare fill does not contribute significantly toM _g . We choose the magnetizing dipole momentM _o and the constantc such that the magnitude of the productcH _o ≃ 3 × 10⁻⁴Г at the outer surface of the shell in the equatorial plane of the dipole. This value of the intensity of remanent magnetization was chosen to be within the range 10⁻⁷−10⁻³Г’; the intensities of thermo-remanent magnetization exhibited by Apollo samples. Finally, we use the locations and diameters of the 10 largest craters on the Moon and the depth-to-diameter ratios of Pike’s formulation to model approximately the excavation of the magnetized shell. The remanent dipole momentM _g was calculated for each of three orthogonal orientations of the magnetizing dipoleM _o. The three magnitudes ofM _g fall in the range 4 × 10¹⁸−1 × 10¹⁹Г cm³ which is close to the upper limit of 10¹⁹Г cm³ estimated forM _g from the field measurements obtained with the Apollo subsatellites. Further, the distribution of the craters is such as to produce a significant transverse component ofM _g with acute angles between the spin axis andM _g in the range 51°–77°.     

Maximum temperatures for radiation from plasma waves

We derive upper limits to the radiation temperaturesT ^t(k) for emission near the fundamental and second harmonic of the electron plasma frequency in terms of the effective temperature for plasma wavesT ^l(k). We findT ^t(k)(c/(3)^1/2 V _e)³ T ^l(k) for emission near the fundamental which differs from the result of Melrose (1970b) by the factor in parentheses. This factor can exceed 4×10⁴ in some plasmas. The conditions under which this limit could be reached are delinated. For emission near the second harmonicT ^t(k)T ^l(k) since the absorption coefficient in this case can only be positive.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

J-Type Shock Chemistry for the W3 IRS5 Region

A chemical evolution of W3 IRS5 have been studied during J-type shock. We modified our code of for one-dimensional J-type shock to satisfy the physical conditions of W3 IRS5. The chemical rate equations and the hydrodynamics of shocks are integrated simultaneously. We find that, a shock of speed V _s = 10 km s^-1 and magnetic induction B=4.1×10^-6 Gauss, propagate into molecular gas density of n=10⁴cm^-3, and with initial temperature of 30 K (pre shock) satisfy the physical conditions of W3 IRS5. The results using the extensive chemical network with the most updating reaction rates show that the calculated fractional abundances are in agreement with observation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Electromagnetic fields satisfying the condition B ∧ (∇ ∧ B)=0

It is proved that (1) electromagnetic fields with electric and magnetic components parallel to one another are solutions of Maxwell's equations; (2) the equationB(B)=0 (B is the magnetic field) is gauge and relativistically-invariant for systems of reference moving with velocityv/c=EB(1+v ²/c ²)/(E ²+B ²).     

An Analytical Model to Predict the Arrival Time of Interplanetary CMEs

Referring to the aerodynamic drag force, we present an analytical model to predict the arrival time of coronal mass ejections (CMEs). All related calculations are based on the expression for the deceleration of fast CMEs in the interplanetary medium (ICMEs), [(v)\dot]=-\frac115 700(v-V_SW)²\dot{v}=-\frac{1}{15\,700}(v-V_{\mathrm{SW}})^{2} , where V _SW is the solar wind speed. The results can reproduce well the observations of three typical parameters: the initial speed of the CME, the speed of the ICME at 1 AU and the transit time. Our simple model reveals that the drag acceleration should be really the essential feature of the interplanetary motion of CMEs, as suggested by Vršnak and Gopalswamy (J. Geophys. Res. 107, 1019, 2002).     

Drift in toroidal configurations

It is found that charged particles of positive energiesE, when constrained on axisymmetric isoflux surfaces , execute sinusoidal motions with typical frequencies =(2E/m)^1/2). In general, it was found that under equilibrium condition p=J ^B/cthe particles develop a non-ambipolar drift velocityv _d =(cµ/eb)[1+q ² +2(q/)²]p.     

The Correlation between the Magnetic and Velocity Fields on the Full Solar Disk

A possible correlation between the magnetic and velocity fields has been analyzed based on the SOHO/MDI magnetograms and Dopplergrams. It is found that the observed large-scale weak magnetic field (weaker than 50 G (gauss)) is correlated with the velocity statistically. The curves of u⋅b with latitude, where u and b are the velocity and magnetic fields in a rectangular region (±15^○ in longitude, ±45^○ in latitude) on the Sun, show the same patterns in the years 2000, 2004, and 2007. The patterns indicate that u and b are positively correlated near the equator but are anti-correlated at the middle latitudes. For a strong magnetic field between 50 G and 3000 G, the curves of u⋅b with latitude show the same tendencies at the middle latitudes. Near the equator, however, the slope of the curve is positive in 2000 and is negative in 2004 and 2007. In addition, we give an estimation for the amplitude of the cross helicity h _χ (h_c=[`(u·b)]h_{\chi}=\overline{\mathbf{u}\cdot\mathbf{b}}) inferred from the MDI data, which is of the order of 10³ G m s⁻¹ near the center of the solar disk.     

Do Long-Lived Features Really Exist in the Solar Photosphere? II. Contrast of Time-Averaged Granulation Images

The decrease in the rms contrast of time-averaged images with the averaging time is compared between four data sets: (1) a series of solar granulation images recorded at La Palma in 1993, (2) a series of artificial granulation images obtained in numerical simulations by Rieutord et al. (Nuovo Cimento 25, 523, 2002), (3) a similar series computed by Steffen and his colleagues (see Wedemeyer et al. in Astron. Astrophys. 44, 1121, 2004), (4) a random field with some parameters typical of the granulation, constructed by Rast (Astron. Astrophys. 392, L13, 2002). In addition, (5) a sequence of images was obtained from real granulation images by using a temporal and spatial shuffling procedure, and the contrast of the average of n images from this sequence as a function of n is analysed. The series (1) of real granulation images exhibits a considerably slower contrast decrease than do both the series (3) of simulated granulation images and the series (4) of random fields. Starting from some relatively short averaging times t, the behaviour of the contrast in series (3) and (4) resembles the t ^−1/2 statistical law, whereas the shuffled series (5) obeys the n ^−1/2 law from n=2 on. Series (2) demonstrates a peculiarly slow decline of contrast, which could be attributed to particular properties of the boundary conditions used in the simulations. Comparisons between the analysed contrast-variation laws indicate quite definitely that the brightness field of solar granulation contains a long-lived component, which could be associated with locally persistent dark intergranular holes and/or with the presence of quasi-regular structures. The suggestion that the random field (4) successfully reproduces the contrast-variation law for the real granulation (Rast in Astron. Astrophys. 392, L13, 2002) can be dismissed.     

Note on the Generalized Hansen and Laplace Coefficients

Recently, Breiter et al. [Celest. Mech. Dyn. Astron., 2004, 88, 153–161] reported the computation of Hansen coefficients X _k ^{γ ,m} for non-integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b _s ^(m), and generalized Laplace coefficients b _s,r ^(m) [Laskar and Robutel, 1995, Celest. Mech. Dyn. Astron., 62, 193–217] that do not require s,r to be integers. In particular, the coefficients X ₀ ^{γ ,m} have very simple expressions in terms of the usual Laplace coefficients b _{γ +2} ^(m), and all their properties derive easily from the known properties of the Laplace coefficients.     

Instability of LBV stars against radial oscillations

Hydrodynamic calculations of nonlinear radial oscillations of LBV stars with effective temperatures 1.5 × 10⁴ K ⩽ T _eff ⩽ 3 × 10⁴ K and luminosities 1.2 × 10⁶ L _⊙ ⩽ L ⩽ 1.9 × 10⁶ L _⊙ have been performed. Models for the evolutionary sequences of Population I stars (X = 0.7, Z = 0.02) with initial masses 70M _⊙ ⩽ M _ZAMS ⩽ 90M _⊙ at the initial helium burning stage have been used as the initial conditions. The radial oscillations develop on a dynamical time scale and are nonlinear traveling waves propagating from the core boundary to the stellar surface. The amplitude of the velocity variations for the outer layers is several hundred km s⁻¹, while the bolometric magnitude variations are within ΔM _bol ⩽ 0_· ^m 2. The onset of oscillations is not related to the κ-mechanism and is attributable to the instability of a self-gravitating envelope gas whose adiabatic index is close to its critical value of Γ₁ = 4/3 due to the dominant contribution of radiation in the internal energy and pressure. The interval of magnitude variation periods (6 days ≤ II ≤ 31 days) encompasses all currently available estimates of the microvariability periods for LBV stars, suggesting that this type of nonstationarity is pulsational in origin.     

Simulation of compact groups of galaxies

Self-consistent simulations of seven groups of galaxies with halos have been performed to find a constraint upon the size of missing halos around spiral galaxies. An initial galaxy, which consists of 100 superstars, has half-mass radius 41 kpc and central velocity dispersion 235 km s^–1. The simulations start from the epoch of maximum expansion. The initial conditions involve a variety of spatial distributions of galaxies, and the velocity dispersion of galaxies as would be permitted for maximum expansion. Dense groups having collapse times shorter than (2/3)H ₀ ^–1 are shown to form multiple mergers in a Hubble timeH ₀ ^–1 . From a comparison of the frequencies of cD galaxies, or multiple mergers, in observed and simulated groups, it is concluded that the effective radius of missing halos is less than 41 kpc.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

The physics of E × B-drifting jets

E × B-drifting jets have been generally ignored for the past 25 years even though they may well describe all the astrophysical jet sources, both on galactic and stellar scales. Here we present closed-form solutions for their joint field-and-particle distribution, argue that the observed jets are near equipartition, with extremely relativistic, monoenergetic e^±-pairs of bulk Lorentz factor γ ≲ 10⁴, and are first-order stable. We describe plausible mechanisms for the jets’ (i) formation, (ii) propagation, and (iii) termination. Wherever a beam meets with resistance, its frozen-in Poynting flux transforms the delta-shaped energy distribution of the pairs into an almost white power law,E ² N _E ∼E ^−∫ with ∫ ≳ 0, via single-step falls through the huge convected potential.     

Fe XXI emission line ratios as electron temperature diagnostics for the coronae of cool stars

A method for the reconstruction of the linear force-free magnetic field in a bounded domain (as a rectangular box, ) is presented here. The Dirichlet boundary-value problem for the Helmholtz equation is solved for the B _z component specified at the boundary. Chebyshev's iteration method with the optimal rearrangement of the iteration parameters sequence was used. The solution is obtained as for the positive-definite, and for the non-sign-definite difference analogue of the differential operator ² u + ² u. Specifying two scalar functions, B _x and B _y on the intersection of the lateral part of the boundary with one selected plane z = constant, and using B _z inside the , we have found B _x and B _y throughout .The algorithm was tested with the numerical procedure which gives the analytic solution B of the linear force-free field (LFFF) equations for the dipole in a half-space. The root-mean-square deviation of the analytic solution B from the calculated B does not exceed 1.0%. Boundary conditions for the B calculation were taken as given by the analytic LFFF solution B. Comparison of B with B, which was calculated by the potential non-photospheric boundary conditions, show that they differ significantly. Thus, the specification of boundary conditions at non-photospheric boundaries of the volume under consideration is of particular importance when modeling the LFFF in a bounded volume.The algorithm proposed here allows one to use the information about magnetic fields in the corona for the modeling of LFFF in a limited domain above an active region on the Sun.     

Radial pulsations of massive main-sequence stars

The evolution of Population I stars (X = 0.7, Z = 0.02) with initial masses 40M _⊙ ≤ M _ZAMS ≤ 120M _⊙ until core hydrogen exhaustion has been computed. Models of evolutionary sequences have been used as the initial conditions in solving the equations of radiation hydrodynamics that describe the spherically symmetric motion of a self-gravitating gas. Stars with initial masses M _ZAMS ≥ 50M _⊙ are shown to become unstable against radial oscillations during the main-sequence evolution. The instability growth rate and the limit-cycle oscillation amplitude increase as the star evolves and as its initial mass increases. The pulsational instability is attributable to the iron Z-bump κ mechanism (T ∼ 2 × 10⁵ K). Convection that transfers from 20 to 50% of the total energy flux and, thus, reduces the efficiency of the κ mechanism emerges in the same layers. The periods of the radial oscillations of main-sequence stars lie within the range from 0.09 to 8 days. The boundaries of the instability region of radial pulsations in the Hertzsprung-Russell diagram have been determined and observational criteria for revealing pulsating variable main-sequence stars have been proposed.     

On the presence of electric currents in the solar atmosphere

An elementary analysis based on Ampére's Law is given to separate the general magnetic field above the photosphere into two parts B=B ₁+B ^*. The field B ₁ is a potential field due to electric currents below the photosphere. The field B ^* is produced by electric currents above the photosphere combined with the induced mirror currents. By symmetry, B ^* has a set of field lines lying entirely in the plane z = 0 taken to be the photosphere. This set of field lines can be constructed from given vector magnetograph measurements and represents all the information on the electric currents above the photosphere that a magnetograph can provide. Theoretical illustrations are given and the implications for data analysis are discussed.     

Neutrino radiation of the AGN black holes

In the framework of ‘microscopic’ theory of black holes (J. Phys. Soc. Jpn. Suppl. B 70, 84, 2001; Astrophys. USSR 4, 659, 1996; 35, 335, 1991, 33, 143, 1990, 31, 345, 1989a; Astrophys. Space Sci. 1, 1992; Dokl. Akad. Nauk USSR 309, 97, 1989b), and references therein, we address the ‘pre-radiation time’ (PRT) of neutrinos from black holes, which implies the lapse of time from black hole’s birth till radiation of an extremely high energy neutrinos. For post-PRT lifetime, the black hole no longer holds as a region of spacetime that cannot communicate with the external universe. We study main features of spherical accretion onto central BH and infer a mass accretion rate onto it, and, further, calculate the resulting PRT versus bolometric luminosity due to accretion onto black hole. We estimate the PRTs of AGN black holes, with the well-determined masses and bolometric luminosities, collected from the literature by Woo Jong-Hak and Urry (Astrophys. J. 579, 530, 2002) on which this paper is partially based. The simulations for the black holes of masses M _BH ≃(1.1⋅10⁶ ÷4.2⋅10⁹) M _⊙ give the values of PRTs varying in the range of about T _BH ≃(4.3⋅10⁵ ÷5.6⋅10¹¹) yr. The derived PRTs for the 60 AGN black holes are longer than the age of the universe (∼13.7 Gyr) favored today. At present, some of remaining 174 BHs may radiate neutrinos. However, these results would be underestimated if the reservoir of gas for accretion in the galaxy center is quite modest, and no obvious way to feed the BHs with substantial accretion.     

Theoretical intensities of Fexiv in the solar EUV spectrum

Equilibrium population of Fexiv levels in coronal conditions was calculated including configurations 3s ²3p, 3s3p ², 3s ²3d, 3p ³, 3s3p3d, 3s ²4s, 3s ²4p, 3s ²4d, 3s ²4f. Relative populations of selected levels are given in Table VII. Figure 1 shows the dependence of relative intensities of the strongest lines on electron density. Certain line ratios can be used for the determination of N _e .E.g., at T=2 × 10⁶ K and with a dilution factor 0.4, the intensity ratio of λ211.3 and λ219.0 changes by a factor of 65 if N _e increases from 10⁷ to 10¹¹ (Table VIII). Cascades from the 3s3p3d and 3p ³ configurations are important in the population of some levels of 3s3p ² (Table VI). A possibility of identification of additional lines in the solar spectrum is indicated. NAS-NRC Resident Research Associate.     

Magnetofrictional Extrapolations of Low and Lou’s Force-Free Equilibria

We present a careful investigation of the magnetofrictional relaxation and extrapolation technique applied to the reconstruction of two test fields. These fields are taken from the family of nonlinear force-free magnetic equilibria constructed by Low and Lou (Astrophys. J. 352, 343, 1990), which have emerged as standard tests for extrapolation techniques in recent years. For the practically relevant case that only the field values in the bottom plane of the considered volume (vector magnetogram) are used as input information (i.e., not including the knowledge about the test field at the side and top boundaries), the test field is reconstructed to a higher accuracy than obtained previously. Detailed diagnostics of the reconstruction accuracy show that the implementation of fourth-order spatial discretization was essential to reach this accuracy for the given test fields and to achieve near machine precision in satisfying the solenoidal condition. Different variants of boundary conditions are tested, which all yield comparable accuracy. In its present implementation, the technique yields a scaling of computing time with total number of grid points only slightly below N ^5/3, which is too steep for applications to large (≥1024²) magnetograms, except on supercomputers. Directions for improvement are outlined.