Crossing of Various Cantori

We find the form of cantori surrounding an island of stable motion in the standard map for various values of the nonlinearity parameter K near the value K=5 (much larger than the critical value K _cr=0.971635...). The asymptotic curves of unstable periodic orbits inside the cantorus cross it after a certain time and then escape to the large chaotic sea. For K=5 the crossing time (in appropriate units) is t=1 and the escape time is t=2. For K=4.998 the crossing time is t=7 and the escape time t=23000. This delay of escape is due to the existence of higher order cantori, with very small gaps. We found that, as K increases the noble torus [2,4,1,1,..] is destroyed before the destruction of the higher order tori [2,4,1,1,1,1,2,1,...] and [2,4,1,1,1,1,3,1,...]. Thus the torus with the simplest noble number is not the last KAM curve to be destroyed. Then we find that nearby orbits deviate considerably, but the average times spent near various resonance before escape are very similar.     

Coronal and interplanetary transport of solar energetic protons and electrons

We present a new method to separate interplanetary and coronal propagation, starting from intensity variations observed by spaceprobes at different heliolongitudes. In general, a decrease in absolute intensities is observed simultaneously with an increase in temporal delays. The coupling of these two effects can be described by Reid's model of coronal diffusion and can in principle be used to determine the two coronal time constants, diffusion time t _c and escape time A. In addition, a least-squares fit method is used to determine the parameters of interplanetary transport, assuming a radial dependence as (r) = ₀(r/1 AU)^b. The method is applied to the two solar events of 27 December, 1977 and 1 January, 1978 which were observed by the spaceprobes Helios 1, Helios 2, and Prognoz 6. Energetic particle data are analysed for 13–27 MeV protons and -0.5 MeV electrons. For the regions in space encountered during these events the mean free path of electrons is smaller than that of protons. Straight interpolation between the two rigidities leads to a rather flat rigidity dependence (P) P ⁿ with n = 0.17–0.25. This contradicts the prediction of a constant mean free path or of the transition to scatter-free propagation below about 100 MV rigidity. In three of the four cases the mean free path of 13–27 MeV protons is of the order 0.17 AU, the mean free path of electrons of the order 0.06 AU. For protons we find b - 0.7 for the exponent of the radial variation.The concept of two different coronal propagation regimes is confirmed. It is remarkable that in both regimes electrons are transported more efficiently than protons. This holds for the temporal delay as well as for the amplitude decrease. This is in contrast with the long existing concept of rigidity independent transport and puts severe limits to any model of coronal transport. For the December event all three spaceprobes are in the fast propagation regime up to an angular distance of 62°. For protons we find a finite delay even in the fast propagation region, corresponding to a coronal delay rate of about 0.8 hr rad^-1 up to 60° angular distance. In contrast, relativistic electrons may reach this distance within a few minutes.The fast transport of electrons and the different behaviour of electrons and protons is in contradiction to the expanding bottle concept. An explanation of coronal transport by shock acceleration directly on open field lines could in principle work in case of protons in the fast propagation region, but would fail in case of the electrons. The fast and efficient transport of electrons is most likely due to a region of field lines extending over a wide range of longitudes directly from the active region into interplanetary space. The much slower transport of both particle types at large azimuthal distances can neither be explained by direct access to open field lines not by the direct shock acceleration concept. A possible explanation is the loop reconnection model in a modified version, allowing for a faster lateral transport of electrons.Now at AEG, 2000 Wedel, F.R.G.     

Variable radio echoes from a jet in QSO 0957+561

Time delay determinations in astrophysics are used most often to find time delays between flux density variations of different spectral bands and/or spectral lines in AGNs and different images of gravitationally lensed QSOs. Here we consider a new algorithm for a complex case, when the time delay is itself a linear function of time and the intensity of echo response is power function of the delay. We apply this method to investigate optical-to-radio delay in the double quasar 0957+561, which is a generally accepted case of gravitational lensing.Radio-optical correlation in QSO 0957+561 was first reported by Oknyanskij and Beskin (1993, hereafter OB) on the basis of radio observations made in the years 1979 to 1990. OB used an idea to take into account the known gravitational lensing time delay to get combined radio and optical light curves and then to use them for determination of the possible radio-from-optical time delay. It was found this way that radio variations (5 MHz) followed optical ones by about 6.4 years with high level of correlation (0.87). Using new radio data (Haarsmaet al., 1996), for the interval 1979–1994 we find nearly the same value for the optical-to-radio delay as has been found before. Additionally we suspect that the time delay value is linearly increasing at about 110 days per year while the portion of reradiated flux in the radioresponse is decreasing.Obtained results indicate that the optical and radio emitting regions are physically related, but have distinct size scales, locations and possibly radiation mechanisms. We conclude that the results can be explained by simple model were the variable radio source is ejected from the central part of the QSO compact component and that the changing time delay between the optical and radio light curves is consequence of light travel effect.     

Collisionless deceleration of fast electron streams in the solar coronal plasma

The collisionless deceleration of electron streams responsible for type IIIb bursts has been investigated. For this the difference between the mean velocities of electron streams at plasma levels corresponding to 25 and 12.5 MHz, on one hand, at 12.5 and 6.25 MHz, on the other hand, is estimated. The mean velocity of electron streams between these levels is determined by the time delay in the moments of arrival of radio bursts from these levels. The distance between plasma levels is determined under the assumption that the (statistical) mean velocity of sources of the diffusive type III bursts is constant and equal toc/3 at all considered levels of the solar corona.It is shown that under this assumption the electron streams with the initial velocities of the order of 0.4–0.8c undergo a sufficient deceleration which is characterized by a decrease in their mean velocity by 15–17% between plasma levels at 25 to 6.25 MHz. The stream deceleration becomes more essential with the growth of the initial velocity of the stream. On the other hand, the deceleration disappears when the initial velocity of the stream is of the order of 0.35c. This critical velocityV _s ^* - 0.35c is assumed to define a boundary between two different expansion regimes of fast electrons moving in the solar corona. In the first regime (V _s >V _s ^* ) the induced scattering of plasma waves produces energy losses of the streams. A decrease in the velocities of streams up to the value of the order of 0.35c is due to these losses. In the second regime (V _s -V _s ^* ) a quasilinear expansion of streams is realized. In this case the energy losses of the streams are almost absent.     

Time-dependent radiation transfer and a possible explanation of the interpulse in CP 0950

The time-dependent equation of radiative transfer is solved exactly and in then-th Gaussian approximation.The atmosphere is plane-parallel and semi-infinite; isotropic scattering is assumed, but the boundary condition at =0 is arbitrary.The results are used to investigate a suggested mechanism for the origin of the secondary pulses in CP 0950; it is found that a binary system of neutron stars can indeed explain formation, time delay and intensity of the observed interpulse.     

Decimetric type III radio bursts and associated hard X-ray spikes

A detailed comparison is made between hard X-ray spikes and decimetric type III radio bursts for a relatively weak solar flare on 1981 August 6 at 10: 32 UT. The hard X-ray observations were made at energies above 30 keV with the Hard X-Ray Burst Spectrometer on the Solar Maximum Mission and with a balloon-born coarse-imaging spectrometer from Frascati, Italy. The radio data were obtained in the frequency range from 100 to 1000 MHz with the analog and digital instruments from Zürich, Switzerland. All the data sets have a time resolution of 0.1 s or better. The dynamic radio spectrum shows many fast drift type III radio bursts with both normal and reverse slope, while the X-ray time profile contains many well resolved short spikes with durations of 1 s. Some of the X-ray spikes appear to be associated in time with reverse-slop bursts suggesting either that the electron beams producing the radio bursts contain two or three orders of magnitude more fast electrons than has previously been assumed or that the electron beams can trigger or occur in coincidence with the acceleration of additional electrons. One case is presented in which a normal slope radio burst at 600 MHz occurs in coincidence with the peak of an X-ray spike to within 0.1 s. If the coincidence is not merely accidental and if it is meaningful to compare peak times, then the short delay would indicate that the radio signal was at the harmonic and that the electrons producing the radio burst were accelerated at an altitude of 4 × 10⁹ cm. Such a short delay is inconsistent with models invoking cross-field drifts to produce the electron beams that generate type III bursts but it supports the model incorporating a MASER proposed by Sprangle and Vlahos (1983).     

Hurst analysis of Mt. Wilson rotation measurements

I study the temporal variation of the solar rotation on time scales shorter than the 11-year cycle by analyzing the daily Mt. Wilson Doppler measurements from 1967 to 1992. The differential rotation is represented by the three coefficients,A, B, andC, of the following expansion: =A +B sin²() +C sin⁴(). TheA, B, andC time series show clearly the 11-year solar cycle and they also show high-frequency fluctuations. The Hurst analysis of these time series shows that a Gaussian random process such as observational noise can only account for fluctuations on time scales shorter than 20 days. For time scales from 20 days to 11 years, the variations of A give rise to a Hurst exponent ofH = 0.83, i.e., the variations ofA are persistent. The temporal variations ofB show the same behavior asC, which is different fromA. From one to 11 years, theB andC variations are dominated by the 11-year cycle, while for time lags shorter than about 250 days, theB andC fluctuations give rise to a Hurst exponent ofH = 0.66, which lies betweenH = 1/2, of a Gaussian random process, and the exponent of the persistent process shown byA. An analysis of the equivalent coefficients of the first three even Legendre polynomials, computed usingA, B, andC, provides additional information. For time scales between 100 and 1000 days, the ranges,R/S, of Legendre polynomial coefficients decrease with increasing order of the polynomials which suggests that the persistent process operates mainly on large spatial scales. The Hurst exponent ofH = 0.83 for variations inA is the same asH for monthly sunspot numbers with time scales between 6 months and 200 years and for¹⁴C radiocarbon data with time scales between 120 years and 3000 years, previously analyzed by other authors. The combined results imply that the underlying solar process shows the same persistent behavior for time scales as short as about 20 days up to time scales of a few thousand years.Operated by the Association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.     

Vibrational stability of stars in thermal imbalance: A solution in terms of asymptotic expansions

Isentropic oscillations of a star in thermal imbalance are defined as those for which, at every istant, the entropy of each mass element of the configuration in the perturbed motion is equal to that of the same mass element in the unperturbed motion.The solution of the equations describing such isentropic oscillations and written in terms of the infinitesimal displacement r(r ₀,t) is presented in terms of asymptotic expansions up to the first order in the parameter /t _s where is the adiabatic pulsation period for the fundamental mode andt _s , a slow time scale of the order of the Kelvin-Helmholtz time.The solution obtained allows one to define, unambiguously, an isentropic part to the coefficient of vibrational stability of arbitrary stellar models in thermal imbalance, as well as to derive a general formula relating the results of a stability analysis in terms of r and r/r.Application of this general solution to the simple case of homologous motion is also given.     

Simple derivation of Symplectic Integrators with First Order Correctors

In this paper we consider almost integrable systems for which we show that there is a direct connection between symplectic methods and conventional numerical integration schemes. This enables us to construct several symplectic schemes of varying order. We further show that the symplectic correctors, which formally remove all errors of first order in the perturbation, are directly related to the Euler—McLaurin summation formula. Thus we can construct correctors for these higher order symplectic schemes. Using this formalism we derive the Wisdom—Holman midpoint scheme with corrector and correctors for higher order schemes. We then show that for the same amount of computation we can devise a scheme which is of order O(h ⁶)+(² h ²), where is the order of perturbation and h the stepsize. Inclusion of a modified potential further reduces the error to O(h ⁶)+(² h ⁴).This revised version was published online in October 2005 with corrections to the Cover Date.     

The origin of gravity

It has been shown by Atkinson (1965) that there is a rigorously exact euclidean interpretation of the general relativity field equations if certain arbitrary definitions of mass (m0 and the velocity of light (c) are invoked. With a preferred (euclidean) frame postulatedab initio, a particularly simple explanation in terms of classical physics may be found for very similar definitions ofm andc. It is not unexpected that with this scheme, all the usual tests of general relativity (light deflexion, perihelion motion, gravitational redshift, and radar delay time) are immediately satisfield. The preferred frame is however identified with a real aether and this requires a return to the Lorentzian interpretation of the special relativistic transformations of space and time variables. It is shown that gravity may be attributed to the action of a temperature gradient in the aether and an explanation of its origin in terms of an ideal relativistic gas is proposed. The temperature gradients are thermodynamically stable and do not diffuse if the relativistic aether ( _A ) is effectively adiabatic and matter is fundamentally a species of aether with instantaneous motion at high (> _A ) relative to the aethereal reast frame. To be consistent with such a picture, it is necessary to assume aether particles are capable of forming temporary associations (not recognized as matter) which take on some of the properties of crystalline solids and thereby become the means of transmitting electromagnetic radiation through space. The aether is essentially treated as a virtually incompressible fluid in which the pressure at any point arises from both random (temperature) and bulk (high ) motions. A number of specific predictions arising from this theory of gravity are indicated and these may serve to discriminate it from general relativity.     

Determination of mass-loss rates from the first order momentW 1 of unsaturated P Cygni line profiles

The escape probability method introduced by Sobolev to treat the transfer of line photons is used in order to derive the expressions of thenth order momentW _n (E()/E _c –1)(–₁₂) ⁿ ·d of a P Cygni profile formed in rapidly expanding envelopes around a central point-like source under various physical and geometrical conditions.With the only assumption that there is mass-conservation of the species in the flow, we state for the case of optically thin lines that the relation between the first order momentW ₁ and the quantityM _n (level), first established by Castoret al. (1981) under more restrictive conditions, is in fact independent of the type of velocity fieldv(r) and a fortiori of the distribution adopted for the radial opacity ₁₂ ^r (X). These results also remain unchanged when including collisions (0) and/or an additional rotational velocity fieldv (r) in the expanding atmosphere. We investigate the presence of an underlying photospheric absorption line and conclude that for realistic cases, neglecting this boundary condition to the radiative transfer leads to an underestimate of the mass-loss rate by a factor of about 20%. By means of a three-level atom model, we demonstrate that all results derived for a single line transition equally apply for an unresolved doublet profile providedW ₁ andM _n (level) are calculated with the weighted wavelength _D and total oscillator strengthf _D of the doublet.Considering the occultation and inclination effects caused by the finite size of the central core, we refine the value of the multiplicative constant fixing the ratio ofW ₁ toM _n (level). We show that this relation allows a determination of the mass-loss rate with an uncertainly less than 30%, irrespective of the sizeL _max of the atmosphere and of the limb-darkening affecting the stellar core. Reviewing all possible sources of error, we finally conclude that this method of deriving a mass-loss rate from the analysis of an unsaturated P Cygni profile is very powerful. The total uncertainty affecting the determination ofM _n (level) from the measurement ofW ₁ should be smaller than 60%.     

Unsteady MHD flow past a porous plate with step function change in suction at slip flow regime

Unsteady two-dimensional hydromagnetic flow of an electrically conducting viscous incompressible fluid past a semi-infinite porous flat plate with step function change in suction velocity is studied allowing a first order velocity slip at the boundary condition. The solution of the problem is obtained in closed form and the results are discussed with the aid of graphs for various parameters entering in the problem.Notations B intensity of magnetic field - H magnetic field parameter,H=(M+1/4)^1/2–1/2 - h rarefaction parameter - L ₁ slip coefficient; ;I, mean free path of gas molecules;f, Maxwell's reflection coefficient - M magnetic field parameter - r suction parameter - t time - t dimensionless time - u velocity of the fluid - u dimensionless velocity of the fluid - U velocity of the fluid at infinity - v suction velocity - v ₁ suction velocity att<=0 - v ₂ suction velocity att>0 - x distance parallel to the plate - y distance normal to the plate - y nondimensional distance normal to the plate - v kinematic viscosity - electric conductivity of the fluid - density of the fluid - shear stress at the wall - nondimensional shear stress at the wall - erf error function - erfc complementary error function     

Slowly rotating cosmological viscous fluid Universe

The perturbation by a spherical rotating shell is investigated in a homogeneous and isotropic cosmological model of viscous fluid distribution to first order in angular velocity (r, t) of matter and the metric rotation function (r, t) which is uniform and non-uniform the exact solutions for (r, t) are obtained for all cosmological models. The physical properties of these solutions are discussed.     

Force-free magnetic fields may have a "loss of equilibrium"

By comparing the light curves in optical, hard x-ray, and soft x-ray wavelengths for 8 well-observed flares, we confirm previous results indicating that the white light flare (WLF) is associated with the flare impulsive phase. The WLF emission peaks within secondsafter the associated hard x-ray peak, and nearly two minutesbefore the 1–8 soft x-ray peak. It is further shown that the peak power in nonthermal electrons above 50 keV is typically an order of magnitudelarger, and the power in 1–8 soft x-rays radiated over 2 strdn at the time of the WLF peak is an order of magnitudesmaller, than the peak WLF power.Operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. Partial support for the National Solar Observation is provided by the USAF under a Memorandum of Undestanding with the NSF.     

Unified iterative methods in orbit determination

The classical problem of Keplerian orbit determination from only three measurements of time and angular coordinates (t _i, _i, _i) has been solved here numerically in two different ways, using Newton's method for non-linear equations in both cases. The first method (Perov, 1989) is based on KS variables, whereas the second emphasizes the fundamental part played by the unified Lambert's equation and the related formulae in that kind of applications. These two methods have been compared and put into practice in various numerical tests based on real asteroid orbits and ficitious Keplerian asteroid, comet and artificial satellite orbits in order to try the stability of these methods for peculiar orbits.     

Galactic tidal perturbations of long-period comets and the distribution of the inclinations

Perturbation of the perihelion distance q of long-period comets by the galactic tidal force is calculated using Cowell's method. It is shown that the maximum perturbation is suffered by those with i (inclination) close to 50 ~ 60 and not by those with i close to 90 , contrary to the prediction of the first order perturbation theory. The dependence of the perturbation of q upon i is compared with the distribution of the inclinations of observed long-period comets and it is shown that the later is not consistent with an isotropic cloud of comets perturbed by the galactic tid alone. A close stellar encounter is unlikely to be an external disturbance. It is argued that giant molecular cloud is the most likely mechanism of the external disturbances.     

Interplanetary acceleration of energetic particles at 1 and 5 AU

Energetic particle (0.1 to 100 MeV protons) acceleration is studied by using high resolution interplanetary magnetic field and plasma measurements at 1 AU (HEOS-2) and at 5 AU (Pioneer 10). Energy changes of a particle population are followed by computing test particle trajectories and the energy changes through the particle interaction with the time varying magnetic field. The results show that considerable particle acceleration takes place throughout the interplanetary medium, both in the corotating interaction regions (CIR) (5 AU), and in quiet regions (1 AU). Although shocks may contribute to acceleration we suggest statistical acceleration within the CIRs is sufficient to explain most energetic particle observations (e.g., McDonaldet al., 1975; Barnes and Simpson, 1976).The first and second order statistical acceleration coefficients which include transit time damping and Alfvén resonance interactions, are found to be well represented byD _T 8.5×10^–6 T ^0.5 MeV s^–1 andD _TT 4×10^–6 T ^1.5 MeV² s^–1 at 5 AU.By comparison, Fisk's estimates (1976), based on quasi-linear theory for transit-time damping, gaveD _TT 5×10^–7 T MeV² s^–1 at 1 AU.     

A complete model of the propagation of solar-flare cosmic rays

A two-stage model of the propagation of 1–50 MeV solar-flare cosmic rays is presented. The first stage consists of a thin spherical shell of radius r _a near the Sun which feeds particles into interplanetary space (the second stage) where they propagate along the Archimedean mean interplanetary magnetic field under the influences of anisotropic diffusion, convection, and energy changes. To calculate the time dependence at a fixed point in space, account is taken of the corotation of flux tubes past the observer.It is shown that the well-known east-west effect of the time-to-maximum cannot be obtained if the injection from the first stage is impulsive and thus a time and longitude dependent release for the second stage is essential. This is achieved by treating the first stage as a thin, spherical, diffusing shell of radius r _a with diffusion coefficient _s, from which particles leak into interplanetary space at a rate determined by the leakage coefficient .With this model we are able to reproduce simultaneously four principal features of solar events observed at r = 1 AU: (i) the east-west effect, i.e. the time-to-maximum as a function of flare longitude; (ii) the three phases of the anisotropy vector variation; (iii) the time-to-convective-phase as a function of flare longitude; and (iv) the longitudinal distribution of the differential intensity. Our best estimates of the parameters of the near-Sun propagation are that 0.01 hr^–1 _s/r _a ² 0.02 hr^–1 and 1/15 hr^–1 1/10 hr^–1. For the interplanetary propagation we estimate /V - 1.2AU with , the effective cosmic-ray diffusion coefficient and V, the solar-wind speed.     

High-energy flare explosions driven by 3-dimensional X-type current loop coalescence

We present a model for high-energy solar flare explosions driven by 3-dimensional X-type current loop coalescence. The 3-dimensional X-type current loop coalescence, where two crossed flux-tubes interact at one point, is a fundamentally new process as compared to the 1-D and 2-D cases studied earlier. This process is studied by a first-order approach of the relevant variables near the point of coalescence; it appears to yield reliable information in a sufficiently large area around this point. It is shown that, following a strong plasma collapse due to the pinch effect, a point-like plasma explosion can be driven while fast magnetosonic shock waves can also be excited. We found that the conditions in the area producing the remarkable flare bursts of 21 May, 1984 were indeed such that the many flare spikes could have been due to 3-D explosive X-type current loop coalescence. We also show, by studying the conditions of shock formation in a gamma ray flare, that the time delay of -rays from the impulsive phase could be the time needed for the shock formation in the flaring region.We draw some general conclusions on the question why certain flares do emit -rays in the MeV energy range, and why other, apparently important and energetic flares, do not. We accentuate the fact that a well-developed high-energy flare has three phases of particle acceleration.     

Type III solar radio burst storms observed at low frequencies

The analysis of a storm of type III solar radio bursts observed in August 1968 between 5 and 0.2 MHz by the RAE-1 satellite has yielded the storm morphology, a possible relation to meter and decameter storms, and an average exciter speed of 0.37 c between 10 and 40 R (Fainberg and Stone, 1970a, b). A continuation of the analysis, based on the apparent dependence of burst drift rate on heliographic longitude of the associated active region, now provides a distance scale between plasma levels in the streamer, an upper limit to the scale size of coronal streamer density inhomogeneities, and an estimate of the solar wind speed. By fixing one level the distance scale is utilized to determine the electron density distribution along the streamer between 10 and 40 R . The streamer density is found to be 16 times that expected for the solar minimum quiet solar wind. An upper limit to the scale size of streamer density inhomogeneities is estimated to be of the order of 1 or 2 solar radii over the same height range. From the progressive delay of the central meridian passage (CMP) of the lower frequency emission, a streamer curvature is inferred which in turn implies an average solar wind speed of 380 km/sec between 14 and 36 R within the streamer.