Regularization of the circular restricted three-body problem using ‘similar’ coordinate systems

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.     

Short-term periodicities of the sun's ‘mean’ and differential rotation

We have looked for periodicities in solar differential rotation on time scales shorter than the 11-year solar cycle through the power- spectrum analysis of the differential rotation parameters determined from Mt. Wilson velocity data (1969–1994) and Greenwich sunspot group data (1879–1976). We represent the differential rotation by a set of Gegenbauer polynomials (()= + (5sin₂–1)+ (21sin₄–14sin₂+1)). For the Mt. Wilson data, we focus on observations obtained after 1981 due to the reduced instrumental noise and have binned the data into intervals of 19 days. We calculated annual averages for the sunspot data to reduce the uncertainty and corrected for outliers occuring during solar cycle minima. The power spectrum of the photospheric mean rotation , determined from the velocity data during 1982–1994, shows peaks at the periods of 6.7–4.4 yr, 2.2 ± 0.4 yr, 1.2 ± 0.2 yr, and 243 ± 10 day with 99.9% confidence level, which are similar to periods found in other indicators of solar activity suggesting that they are of solar origin. However, this result has to be confirmed with other techniques and longer data sets. The 11-yr periodicity is insignificant or absent in . The power spectra of the differential rotation parameters and , determined from the same subset, show only the solar cycle period with a 99.9% confidence level.The time series of determined from the yearly sunspot group data obtained during 1879–1976 is very similar to the corresponding time series of . After correcting for data with large error bars (occurring during cycle minima), we find periods, which are most likely harmonics of the solar cycle, such as 18.3 ± 3.0 yr and 7.5 ± 0.5 yr in and confirmed these and the 3.0 ± 0.1 yr period in . The original time series show in addition some shorter periods, absent in the corrected data, representing temporal variations during cycle minimum. Given their large error bars, it is uncertain whether they represent a solar variation or not. The results presented here show considerable differences in the periodicities of and determined from the velocity data and the spot group data. These differences may be explained by assuming that the rotation rates determined from velocity and sunspot data represent the rotation rates of the Sun's surface layers and of somewhat deeper layers.     

Search for relativistic companions in ‘non-X-ray’ binary systems

Consideration is given to a search for relativistic objects in massive close binary systems without strong X-ray emission (L _x <10³⁴ erg s^–1). It is pointed out that, according to the present-day theory on the evolution of massive close binaries, the number of neutron stars and black holes in non-X-ray binary systems must be 100 times the number of the known X-ray binaries comprising OB supergiant stars; that is why, in studying non-X-ray binary systems, the chances are to detect about a hundred of black holes in the Galaxy.Criteria are formulated for the relativistic nature of companions in the binary systems, such as high spatial velocity values and height Z over the galactic plane for OB stars (runaway stars) and for Wolf-Rayet stars. As reported by Tutukov and Yungelson (1973), as well as by van den Heuvel (1976), the presence of ring-type nebulae can serve as another indication of a relativistic nature of companions in the case of Wolf-Rayet stars.Data are collected on Wolf-Rayet stars with low-mass companions (Table I), which can be relativistic objects accreting within a strong stellar wind from Wolf-Rayet stars. Presented are new findings in respect of spectral examination of the runaway OB-stars (Table II), bringing together data on eight OB stars which can represent binary systems with relativistic companions (Table III).A list of 28 OB-stars (Table IV) which offer a good chance for finding relativistic companions is given.     

Pitch angle scattering of solar particles: Comparison of ‘particle’ and ‘field’ approach

In the quasi-linear theory of pitch angle scattering the power spectrum of magnetic field fluctuations is related to the shape of the pitch angle diffusion coefficient D(), the absolute value of the mean free path , and the rigidity dependence of the mean free path (R). We discuss these relations in detail during the solar particle event of 11 April, 1978 which was observed on HELIOS-2 at a distance of 0.49 AU from the Sun. Magnetic field measurements obtained during the time of the event are used as a basis for the layer model in which the method of particle trajectories in an actually measured field is used to simulate pitch angle diffusion. The values of D() and based on the trajectory simulation for 100 MeV protons (field approach) are compared with results obtained from solar proton data (particle approach) and with predictions from quasi-linear theory based on the additional assumption of the slab model for magnetic field fluctuations (QLT approach). The time of the event is characterized by a high level of field fluctuations, the observed mean free path of about 0.03 AU for 100 MeV protons is smaller than the average value near 1 AU. Results from the field and particle approaches agree surprisingly well. The remaining difference in the mean free path of about a factor of 2 could be due to tangential discontinuities which are measured by the magnetometer, but not seen by the real particles traveling along the average field. The results from the field and QLT approaches based on the same set of magnetic field measurements differ by about a factor of 4. One of the reasons for this discrepancy is that the conditions for resonance scattering are only marginally valid. In addition, the wave vectors representing Alfvén-type fluctuations may not be totally field aligned. This deviation from the slab model would cause an increase of the theoretically predicted mean free path and lead to better agreement with the other two approaches.     

On ‘bimodality of the solar cycle’ and the duration of cycle 21

The period-growth dichotomy of the solar cycle predicts that cycle 21, the present solar cycle, will be of long duration (>133 mo), ending after July 1987. Bimodality of the solar cycle (i.e., cycles being distributed into two groups according to cycle length, based on a comparison to the mean cycle period) is clearly seen in a scatter diagram of descent versus ascent durations. Based on the well-observed cycles 8–20, a linear fit for long-period cycles (being a relatively strong inverse relationship that is significant at the 5% level and having a coefficient of determination r ² 0.66) suggests that cycle 21, having an ascent of 42 mo, will have a descent near 99 mo; thus, cycle duration of about 141 mo is expected. Like cycle 11, cycle 21 occurs on the downward envelope of the sunspot number curve, yet is associated with an upward first difference in amplitude. A comparison of individual cycle, smoothed sunspot number curves for cycles 21 and 11 reveals striking similarity, which suggests that if, indeed, cycle 21 is a long-period cycle, then it too may have an extended tail of sustained, low, smoothed sunspot number, with cycle 22 minimum occurring either in late 1987 or early 1988.     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

Guide and examples for users of the ‘spacekap’ style file

Guide and examples for users of the spacekap style fileBasic instructions     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

On the structure of Saturn's rings and the ‘real’ rotational period for the planet

An analysis of the Lowell Observatory photographic plates of Saturn gave the following results: (1) ring A and B show peculiar brightness distributions around the planet, from which we conclude that both are composed of particles in synchronous rotation. (2) The leading side of the particles in ring A is brighter than the trailing side by about 4%, which may indicate an interaction between such particles and the interplanetary medium. (3) Scans of the rings across the major axis show a small (～0.3″) region of enhanced brightness, from which we derive a value ofT _s =10^h13 _. ^m 8±5 _. ^m 4 for the actual planetary rotational period of Saturn. (4) In order to explain the synchronous rotation, the particles in ring A have to be at least 42 m in diameter.     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

Guide and examples for users of the ‘Spacekap’ style file

Guide and examples for users of the Spacekap style fileBasic Instructions     

Slow-shock heating and the Kopp-Pneuman model for ‘post’-flare loops

The heating of post-flare loops in the Kopp-Pneuman (1976) model is here reconsidered. In that kinematic model the loops are heated by gas-dynamic shocks to at most 3–4 × 10⁶ K. However, in a full dynamic model they would be replaced by slow magnetohydrodynamic shocks, which may provide more heating due to the additional release of magnetic energy. It is shown from a local compressible analysis that such shock waves can account for the observed temperatures of 5 × 10⁶–10⁷ K and also for the observed upward loop speeds of 1–50 km s^-1. The above values are obtained when the ambient plasma beta is 0.01 and the shocks propagate at highly sub-Alfvénic velocities. However, if the velocity of shock propagation approaches the Alfvén speed, then temperatures of 10⁸ K are produced. This may explain the extremely high temperatures that have been observed with the Solar Maximum Mission, when it is realised that the post-flare loop phenomenon may well be occurring very early on in the flare.A full dynamic model would require a sophisticated numerical computation, and so a simple global analytic model is developed here instead. It is incompressible and includes a strong solar-wind inflow along the reconnecting field lines. As the upflow increases, the loops become more compressed and the Alfvén waves approach one another.     

New tortoise coordinate transformation and Hawking’s radiation in de Sitter space

Hawking’s radiation effect of Klein-Gordon equation, Dirac particles and Maxwell’s electromagnetic fields in the non-stationary rotating de Sitter cosmological space-time is investigated by using a new method of generalized tortoise coordinate transformation. It is found that the new transformation produces constant additional terms in the expressions of the surface gravities and the Hawking’s temperatures. If the constant terms are set to zero, then the surface gravities and Hawking’s temperatures will be equal to those obtained from the old generalized tortoise coordinate transformations. This shows that the new transformations are more reasonable. The Fermionic spectrum of Dirac particles displays a new spin-rotation coupling effect.     

Guide and examples for users of the ‘SPACEKAP’ style file

Guide and examples for users of the SPACEKAP style fileBasic instructions     

The origin of the ‘FUN’ anomalies and the high temperature inclusions in the allende meteorite

The discovery of isotopic anomalies in white inclusions of the meteorite Allende has led to fundamental questions concerning the origin of these anomalies and of the white inclusions themselves. An analysis of the FUN anomalies in the inclusions C1 and EK1-4-1 demonstrates that these isotopic anomalies may be decomposed into individual nucleosynthetic components, which have been subjected to separate mass and component fractionations. There is no evidence that any freshlysynthesized material injected into the primitive solar nebula was of abnormal isotopic composition, or that the FUN anomalies were due to an injection of unusual material. Rather, they show the effects of form of interstellar grains whose size or chemistry served as a memory for the nucleosynthetic origins of their constituent atoms. Giant gaseous protoplanets, as described for the early solar nebula by Cameron (1978), are a potential site for achieving both mass and component fractionations, and for producing white inclusions in general.