Non-thermal solar wind heating by supra-thermal ions

The effect of a new energy source due to energies transferred from supra-thermal secondary ions on the temperature profile of the solar wind has been considered. For this purpose a solution of a tri-fluid model of the solar wind including solar electrons, protons, and -particles, and starting with the boundary conditions of Hartle and Barnes at 0.5 AU is given. On the base of the assumption that suprathermal He⁺-ions which have four times the temperature of suprathermal protons are predominantly coupled to solar -particles by Alfvén waves, it is shown that the temperature T of solar -particles should be appreciably higher than those T _p of solar protons beyond the orbit of the Earth. For 1 AU a temperature excess T over T _p according to that which has been found in some solar wind ion spectrograms can only be explained for a small part of the orbit of the earth which is inside the cone of enhanced helium densities. Around 1 AU the temperatures T and T _p are found to decrease much slighter with solar distance than given in the two-fluid model of Hartle and Barnes. Beyond 1.7 and 2.2 AU the temperatures T and T _p even start increasing with solar distance and come up to about 10⁵ at about 10 AU. These predictions should lend some support to future temperature measurements with deep-space probes reaching Solar distances of some AU.Forschungsberichte des Astronomischen Institutes, Bonn, 72-10.     

The motion of a satellite in an axi-symmetric gravitational field

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.List of Symbols semi-major axis - C _jk ⁿ (, ) cosine functions of and - e eccentricity of the orbit - f acceleration vector of perturbing force - f sin²t - i inclination of the orbit - J _n coefficients in the potential expansion - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r radius-vector of satellite - r magnitude ofr - S _jk ⁿ (, ) sine functions of and - T time of perigee passage - u argument of latitude - U gravitational potential - true anomaly - V perturbing potential - G(M₊+m) (gravitational constant times the sum of the masses of Earth and satellite) - _n,k coefficients ofR component of disturbing acceleration (funtions off) - _n,k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^n,m zero-order Hansen coefficients - argument of perigee - right ascension of the ascending node     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

Possible mass limits for quantum double galaxies

When cosmic quantum mechanics is applied to a double galaxy, the result is mass limits in order for the two galaxies to form a quantum binary system. For a non-relativistic theory (based on the Schrödinger wave equation), the mass limits are: (m _g)_max 10¹² M and (m _g)_min 10¹⁰ M . One possible consequence appears to be a Newtonian gravitational constant that varies with cosmic time, with its value larger in the cosmic past.     

Fourier analysis of the light curves of eclipsing variables,XII

The aim of the present paper will be to make use of the expressions, established in Paper XI, for the fractional loss of light _l ⁰ of arbitrarily limb-darkened stars in the form of Hankel transforms of zero order, in order to evaluate the explicit forms of the _l ⁰'s for different types of eclipses (Section 2), as well as of the momentsA _2mof the respective light curves (Section 3)-in a closed form; or in terms of expansions that converge under all circumstances envisaged. Particular attention will be directed to a connection between these expansions and other functions already available in tabular form; or to alternative forms amenable to automatic computation.     

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

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.     

The motion of a satellite in a ring potential

This investigation presents the orbital elements of a satellite moving in a circular ring potential. The ring is considered to be of infinitesimal thickness and of unit radius. The components of the perturbing accelerations due to the ring potential have been substituded into the Gauss form of Lagrange's planetary equations to yield the first-order approximations. The elements of the orbit have been expressed by means of Hansen coefficients. The results include the effects produced by the 2nd, 4th, 6th, and 8th spherical harmonics. Due to their importance we present separately the secular terms from the periodic ones. The general expressions for the orbital elements can be easily extended to include the effects produced by any other higher harmonic.List of Symbols semi-major axis - C _jK ⁿ (u, ) cosine functions ofu and - e eccentricity of the orbit - f sin² - inclination of the orbit - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r magnitude of position vector - S _jK ⁿ (u, ) sine functions ofu and - T time of periapse passage - u argument of latitude - U gravitational potential - V perturbing potential - G(M _r +m) (gravitational constant times the sum of the masses of ring and satellite) - _{n, k} coefficients ofR component of disturbing acceleration (functions off) - _{n, k} coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^{n, m} zero-order Hansen coefficients - argument of periapse - longitude of the ascending node     

Chaos in the solar system

We have accumulated thousands of orbits of test particles in the Solar System from the asteroid belt to beyond the orbit of Neptune. We find that the time for an orbit to make a close encounter with a perturbing planet, T _c ,is a function of the Lyapunov time, T _ty .The relation is log (T _c /T _o )= a + b log (T _ly T _o )where T _o is a fiducial period which we have taken as the period of the principal perturber or the period of the asteroid. There are exceptions to this rule interior to the 2/3 resonance with Jupiter. There, at least in the restricted problem, for sufficiently small Jupiter mass, orbits may have a positive Lyapunov exponent and still be blocked from having a close approach to Jupiter by a zero velocity curve. Of more serious concern is whether the relation holds for purely secular resonances, and if it does, how to choose T _o .This is the case of interest for the planets in the solar system.     

On a transformation of the differential equations of the lunar theory

This work contains a transformation of Hill-Brown differential equations for the coordinates of the satellite to a type which can be integrated in a literal form using an analytical programming language. The differential equation for the parallax of the satellite is also established. Its use facilitates the computation of Hill's periodic intermediary orbit of the satellite and provides a good check for the expansion of the coordinates and frequencies. The knowledge of the expansion of the parallax facilitates the formation of differential equations for terms with a given characteristic. These differential equations are put into a form which favors the solution by means of iteration on the computer. As in the classical theory we obtain the expansions of the coordinates and of the parallax in the form of trigonometric series in four arguments and in powers of the constants of integration. We expand the differential operators into series in squares of the constants of integration. Only the terms of order zero in these expansions are employed in the integration of the differential equations. The remaining terms are responsible for producing the cross-effects between the perturbations of different order. By applying the averaging operator to the right sides of the differential equations we deduce the expansion of the frequencies in powers of squares of the constants of integration.Basic Notations f the gravitational constant - E the mass of the planet - M the mass of the satellite - t dynamical time - x, y, z planetocentric coordinates of the satellite - u x+y–1 - s x–y–1 - the planetocentric distance of the satellite - w 1/ - ₀ the variational part of - w ₀ the variational part ofw, - n the mean daily sidereal motion of the satellite - a the mean semi-major axis of the satellite defined by means of the Kepler relation:a ³ n ²=f(E+M) - a the mean semi-major axis defined as the constant factor attached to the variational solution - e the constant of the eccentricity of the satellite - the sine of one half the orbital inclination of the satellite relative to the orbit of the sun - c(n–n) the anomalistic frequency of the satellite - c ₀ the part ofc independent frome,e, and - g(n–n) the draconitic frequency of the satellite, - g ₀ the part ofg independent frome,e, and - exp (n–n)t–1 - D d/d - e the eccentricity of the solar planetocentric orbit - a the semi-major axis of the solar orbit - n the mean daily motion of the sun in its orbit around the planet - m n/(n–n) - a/a-the parallactic factor - the disturbing function     

GPS-Based On-Board Orbit Determination of a Satellite Using Extended H ∞ Filtering Algorithms

In this paper the extended H filtering algorithms for the design of the GPS-based on-board autonomous navigation system for a low earth orbit (LEO) satellite are introduced. The dynamic process models for the estimation of position, velocity and acceleration from the GPS measurements are established. The nominal orbit of the small LEO satellite is determined by using the 7th–8th order Runge—Kutta algorithms. Three filtering approaches are applied to smooth the orbit solutions, respectively, based upon the simulated GPS pseudo range observables using the Satellite Navigation Tool Box. The simulation shows that the observed orbit errors obtained by using the extended H filtering algorithms can be reduced to a lower level than the observed orbit errors in the sense of RMS within 12 h of tracking time by using the H filtering algorithms and the extended Kalman filtering algorithms under the appropriately designed parameters. Based upon the position errors predicted by the three filtering algorithms after the last observation, we find that the extended H filtering algorithm provides the least position errors of the user satellite.This revised version was published online in October 2005 with corrections to the Cover Date.     

Further considerations on contracting solar nebula

Prentice (1978a) in his modern Laplacian theory of the origin of the solar system has established the scenario of the formation of the solar system on the basis of the usual laws of conservation of mass and angular momentum and the concept of supersonic turbulent convection that he has developed. In this, he finds the ratio of the orbital radii of successively disposed gaseous rings to be a constant - 1.69. This serves to provide a physical understanding of the Titius-Bode law of planetary distances. In an attempt to understand the law in an alternative way, Rawal (1984) starts with the concept of Roche limit. He assumes that during the collapse of the solar nebula, the halts at various radii are brought about by the supersonic turbulent convection developed by Prentice and arrives at the relation: R _p= Ra^p, where R _pare the radii of the solar nebula at various halts during the collapse, R the radius of the present Sun and a = 1.442. a is referred here as the Roche constant. In this context, it is shown here that Kepler's third law of planetary system assumes the form: T _p = T ₀(a^3/2)^p, where T _p are the orbital periods at the radii R _p, T ₀ - 0.1216d - 3 h, and a the Roche constant. We are inclined to interpret T ₀' to be the rotation period of the Sun at the time of its formation when it attained the present radius. It is also shown that the oribital periods T _pcorresponding to the radii R _psubmit themselves to the Laplace's resonance relation.     

Effect of solar radiation on a swarm of meteoric particles

The theory of the Poynting-Robertson effect is applied to the motion of meteors relative to a parent-comet describing an undisturbed elliptic orbit. It is shown that initially any emitted particle proceeds to move retrogressively away from the comet to a certain maximum angular distance (as seen from the Sun) depending on its s-value, and thereafter undergoes relative motion in the opposite forward direction. The time taken to reach this greatest elongation behind the comet is the same for all particles, and after twice this time the particles will have returned to zero angular displacement relative to the comet. As the inward radial displacement is of far smaller order of magnitude, this means that a swarm of particles will come together again simultaneously, and then move on forwards relative to the comet as they are drawn in slowly towards the Sun. For comet Encke the time for the elongation to return to zero is about 6600 y, for Halley it is about 2×10⁵ y, and for Tempel-Tuttle (1965 IV) just over 10⁵ y. Since this last comet is known to have been deflected from a long-period orbit to a short-period orbit in the year 126 A.D., the theory yields an upper limit to the s-values of about 2.3×10^–2 g cm^–2 for such of its particles as have spread right round the orbit to give rise to the annual November Leonids. Also, for the great meteor-storms associated with this comet, the particles are still moving close behind the comet itself, and their s-values must be about 6.2×10^–2 g cm^–2. This result together with their observed brightnesses suggest that the particles have an effective density little more than 0.1 g cm^–3.     

Spin-orbit coupling: A unified theory of orbital and rotational resonances

The dynamics of the spin-orbit interaction of a sphereM ₈ and a rotating asymmetrical rigid bodyM _a are examined. No restrictions are imposed on the masses, on the orientation of the rotation axis to the orbit plane, or on the orbit eccentricity. The zonal potential harmonics ofM _a induce a precession of the spin axis as well as a precession of the orbit plane, the net effect being a uniform precession of the node on an invariant plane normal to the constant total angular momentum of the system. In general, the effect of the tesseral harmonics is to induce short-period perturbations of small amplitude in both the orbital and spin motions. Resonances are shown to exist whenever the orbital and rotational periods are commensurable. In any resonant state a single coordinate is found to represent both orbital and spin perturbations; and the system may be described as trapped in a localized potential well. The resultant spin and orbit librations are in phase with a common period. The relative amplitudes of the spin/orbit modes are determined by the characteristic parameter =M _a M _s a ² /3(M _a +M _s )C, wherea is the semimajor axis of the orbit, andC is the moment of inertia ofM _a about the rotation axis. When ga1, the solutions reduce to those for pureorbital resonance, in whichM _s librates in an appropriate reference frame while the rotation rate of the asymmetrical body remains constant. In the opposite extreme of 1, the solutions are appropriate to purerotational resonance, in which the orbital motion is unperturbed but the spin ofM _a librates. In each of these special cases the equations developed herein on the basis of a single theory are in agreement with those previously determined from separate theories of spin and orbital resonances.     

Asymptotic properties of disk dynamo

The asymptotic properties of a turbulent disk dynamo at large dimensionless numbersR andR characterizing the helicity and the differential rotation are analysed. Three types of generations in the dependence of the relations betweenR andR are found: ²-dynamo and two types of -dynamo. For each of these types the rates of growth are obtained and the forms of solution are pointed out. Boundaries of the disk dynamo approximation are given.     

Flare-like energy release by flux pile-up reconnection

Flux pile-up magnetic merging solutions are discussed using the simple robust arguments of traditional steady-state reconnection theory. These arguments determine a unique scaling for the field strength and thickness of the current layer, namely B _s^–1/3, l^2/3, which are consistent with a variety of plasma inflow conditions. Next we demonstrate that flux pile-up merging can also be understood in terms of exact magnetic annihilation solutions. Although simple annihilation models cannot provide unique reconnection scalings, we show that the previous current sheet scalings derive from an optimized solution in which the peak dynamic and magnetic pressures balance in the reconnection region. The build-up of magnetic field in the current sheet implicit in flux pile-up solutions naturally leads to the idea of saturation. Hydromagnetic pressure effects limit the magnetic field in the sheet, yielding an upper limit on the reconnection rate for such solutions. This rate is still far superior to the Sweet–Parker merging rate, which can be derived by seeking solutions that avoid all forms of saturation. Finally we compare time dependent numerical simulations of the coalescence instability with the optimized flux pile-up models. This comparison suggests that merging driven by the relatively slow approach of large flux systems may be favored in practice.     

Asymptotic solutions in the many-body problem

A small particle moves in the vicinity of two masses, forming a close binary, in orbit about a distant mass. Unique, uniformly valid solutions of this four-body problem are found for motion near both equilateral triangle points of the binary system in terms of a small parameter , where the primaries move in accordance with a uniformly-valid three-body solution. Accuracy is maintained within a constant errorO(⁸), and the solutions are uniformly valid as tends to zero for time intervalsO(^–3). Orbital position errors nearL ₄ andL ₅ of the Earth-Moon system are found to be less than 5% when numerically-generated periodic solutions are used as a standard of comparison.     

Numerical simulations of pulsationally unstable accretion disks around supermassive black holes

The innermost region of slim accretion disks with standard viscosity is unstable against axisymmetric radial inertial acoustic perturbations under certain conditions. Numerical simulations are performed in order to demonstrate behaviors of such unstable disks. It is shown that oscillations with the period of 10^–3 (M _BH/M ) s can be excited near the inner edge of the disks, whereM _BH is the mass of the central object. This kind of unstable disks is a possible origin of the periodic X-ray time variabilities with period of 10⁴s observed in a Seyfert galaxy NGC 6814.     

Evolution of binaries with ultra-short periods: Systematic study

A detailed investigation of the evolution of low-mass binaries is performed for the case when the secondary fills its Roche lobe at the stage of core hydrogen exhaustion. The obtained results are compared with observational data for ultra-short periodic X-ray systems MXB 1820-30 and MXB 1916-05. In the frame of the proposed evolutionary scenario it is possible to obtain for MXB 1820-30 its periodP=11.4 min twice (see Figure 2). In the first case the parameters of the system are:M ₂ 0.13–0.15M ,X0.05–0.13, |P/P| (3.6–6.2) } 10^–7 yr^–1, M₂ (4.1–9.6) } 10^–9 M yr^–1, for the second:M ₂ 0.08–0.09M ,X= 0, |P/P| (1.3–1.5) } 10^–7 yr^–1, M₂ (1.4–1.8) } 10^–8 M yr^–1. It is suggested that MXB 1916-05 is the progenitor of the system MXB 1820-30 (M ₂ = 0.1M,X 0.221,M ₂ 1.8 × 10^–10 M yr^–1).     

Analysis of the light curves of RT Persei by automatized fourier techniques

A new method has been developed by Kopal (1977c, Paper XII) to make use of expressions for the fractional loss of light ₀ ^l of the arbitrarily limb-darkened stars in the form of Hankel transforms of zero-order, in order to evaluate the explicit forms of the ₀ ^l s for different types of eclipse, as well as of the momentsA _2m of the respective light curves in a closed form. The automated method has been tested successfully on the light curves of RT Persei. Also, a photometric curve fit of RT Persei is investigated by application of numerical quadratures to determine the theoretical light curve appropriate for the Roche model. Finally a comparative discussion is given of various methods of light curve analaysis.On leave from Department of Physics, University of Ferdowsi, Mashad, Iran.     

Pulsars: Observational parameters and a discussion on dispersion measures

The relevant data for the known 147 pulsars are presented in graphical and tabular forms. Various data correlations are discussed, and a detailed analysis of pulsar dispersion measures and distances is given. The range of the electron densities in the diffuse interstellar medium is found to be 0.01 cm^–3n _e0.1 cm^–3, and n _e0.03 cm^–3. The dispersion scale height for pulsars is found to be 5.9±0.7 pc cm^–3 implying a linear scale height of 200 pc, which is much smaller than the electron scale height of our Galaxy.Astrophysics and Space Science Review Paper.