On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

Periodic Trojan orbits in the elliptic restricted problem

1966 ◽  
Vol 25 ◽  
pp. 176-186
Author(s):  
E. Rabe
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Restricted Problem ◽  
Angular Motion ◽  
Basic Period ◽  
Short Period ◽  
Elliptic Restricted Problem ◽  
The Mean

As previously shown the generally non-periodic librations about the equilateral points of the plane elliptic restricted problem depend on three frequencies. After elimination of the short-period oscillations, the solutions depend on the libration frequencynand the mean angular motionNof Jupiter. Whenn/Nis commensurable the librations become periodic. One periodic solution corresponds to the actual Trojans, but additional planar solutions exist also with periods equal to multiples of the basic period. For some specific values of the eccentricityethese solutions are accompanied by an associated sequence of non-planar periodic solutions. The values ofedepend slightly on the inclinationi.

Slow variability and circumstellar shells of red variable stars

1987 ◽  
Vol 122 ◽  
pp. 541-542
Author(s):  
T. Lloyd Evans
Keyword(s):  
Variable Stars ◽  
Circumstellar Dust ◽  
Infrared Photometry ◽  
Constant Ratio ◽  
Giant Stars ◽  
Long Period ◽  
Period Variations ◽  
Cyclic Variations ◽  
Circumstellar Shells ◽  
The Mean

Infrared photometry shows that while all RV Tauri stars have circumstellar dust shells, the RVb stars with slow cyclic variations in mean light as well as the 30–100 day variations common to all RV stars have more hot dust close to the star (Lloyd Evans 1985). Many M giant stars which are variables of semiregular type also show long-period variations in the mean light (O'Connell 1933; Payne-Gaposchkin 1954), with a roughly constant ratio between the two periods. Payne-Gaposchkin (1954) found P2/P1 ~9.4 for red variables of type M and P2/P1 ~19.4 for stars of type F-K, most of which are FV Tauri stars. Re-analysis using the more extensive data available now indicates P2/P1 ~10 for the M giants and P2/P1 ~15 for the FV Tauri stars. The nature of the long-period variability is unknown (Wood 1975).

scholarly journals On the Long-Period Variations in the Rate of the Earth's Rotation

1979 ◽  
Vol 82 ◽  
pp. 59-60
Author(s):  
A. I. Emetz ◽  
A. A. Korsun'
Keyword(s):  
Power Spectrum ◽  
Maximum Entropy ◽  
Rotational Speed ◽  
Monthly Data ◽  
Long Period ◽  
Period Variations ◽  
Similar Series ◽  
The Mean ◽  
Using Data ◽  
Linear Trends

The maximum entropy power spectrum (Smylie, et al., 1974) of the Earth's rotational speed was calculated using data from 1900 to 1976. Two series of data were analyzed. The first was a series of δω/ω) determined from annual UT1 - ET data from 1900 to 1976. The second was a similar series derived from the mean monthly data of UT1 - TAI. Linear trends were removed from both series before analysis. Using the second series of data, significant periods of 2.8, 3.7, 7.0, and 10.5 years were found. The first series showed significant periods at 6, 10, 13, 22, and 57 years. Of these periodicities those at 22 and 57 years showed the largest amplitudes (0.454 ± 0.097 × 10−8 and 1.431 ± 0.104 × 10−8 respectively).

scholarly journals Dynamics of the Mimas-Tethys System

1999 ◽  
Vol 172 ◽  
pp. 419-424
Author(s):  
Sylvain Champenois
Keyword(s):  
Long Period ◽  
Numerical Integrations ◽  
The Mean ◽  
Probability Of Capture

AbstractThe role of the 200 yr-long period found recently in the mean longitude of Mimas (Vienne and Duriez 1992) is investigated through numerical integrations. It is shown that it has a deciding effect on the descriptions of the resonance motion of the Mimas-Tethys system, as considered up to now. As a result, Mimas’s inclination before capture may have been higher (up to 0.7°) or lower (down to 0.03°) than the value previously considered (0.42°). Also, Tethys’s eccentricity on capture may have been quite higher (≈ 0.008 versus 0). Moreover, the probability of capture is found to be very sensitive to Tethys’s eccentricity, and possibly much higher (up to 1) than the value considered before (0.04).

scholarly journals A Theory of the Trojan Asteroids

1978 ◽  
Vol 41 ◽  
pp. 145-145
Author(s):  
B. Garfinkel
Keyword(s):  
Periodic Solution ◽  
Restricted Problem ◽  
Elliptic Integral ◽  
Mass Parameter ◽  
Small Mass ◽  
Polar Coordinates ◽  
Lagrangian Point ◽  
Elementary Functions ◽  
The Family ◽  
The Mean

AbstractThe paper constructs a long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. The polar coordinates r and 0 appear in the formHere λ is the mean synodic longitude, m is the small mass-parameter, k is the integer nearest to the ratio ω2/ω1 of the fundamental angular frequencies of the motion, and ck is a Fourier coefficient of a certain periodic function. Only elementary functions enter r(λ) and θ(λ), while the calculation of λ(t) requires the inversion of a hyper-elliptic integral t(λ).The internal resonant terms, carrying the critical divisor D, impart to the orbit an epicyclic character, in qualitative accord with the results of the numerical integration by Deprit and Henrard (1970). Our solution is valid except in the vicinity of the singularities at D = 0 and λ = 0.The presence of the resonant terms invalidates the Brown conjecture (1911) regarding the termination of the family of the tadpoleshaped orbits at the Lagrangian point L3. However, this conjecture holds for the mean orbits defined by r = r(λ), θ = θ(λ), and it also holds in the limit as m → 0.

scholarly journals On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

1978 ◽  
Vol 41 ◽  
pp. 319-323
Author(s):  
P.J. Message ◽  
D.B. Taylor
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Initial Conditions ◽  
Characteristic Exponent ◽  
Mean Value ◽  
Mirror Image ◽  
Mean Values ◽  
Major Semi Axis ◽  
The Mean

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

scholarly journals A Search for Spectroscopic Binaries among Southern Cepheids

1971 ◽  
Vol 15 ◽  
pp. 204-205
Author(s):  
T. Lloyd Evans
Keyword(s):  
Observational Data ◽  
Binary Systems ◽  
Radial Velocities ◽  
Spectroscopic Binaries ◽  
Mean Velocity ◽  
Long Period ◽  
Period Variations ◽  
The Mean

AbstractA discussion of radial velocities determined at the Radcliffe Observatory for 20 southern Cepheids and those found in the literature for another 20 stars lead to the conclusion that 15% of Cepheids are members of long period binary systems (1). The observational data were heterogeneous, few stars having been observed by a single observer or with the same instrument over a sufficient length of time to detect long period variations in the mean velocity.

scholarly journals General Theory for the Outer Planets

1992 ◽  
Vol 152 ◽  
pp. 37-42 ◽  
Author(s):  
P. Bretagnon ◽  
G. Francou
Keyword(s):  
General Theory ◽  
Iterative Method ◽  
Large Amplitude ◽  
Outer Planets ◽  
Long Period ◽  
Fundamental Frequencies ◽  
The Mean ◽  
The Masses ◽  
Planetary Theories

An iterative method for the construction of planetary theories has been developed in order to determine the high order perturbations with respect to the masses. These perturbations are indeed needed to enlarge the validity span of analytical theories up to some million years. The application to the simplified Sun-Jupiter-Saturn problem gives a solution accurate over several ten million years. Throughout the study of the four outer planets we meet with convergence difficulties especially in the determination of fundamental frequencies. One of the results of this study is it shows evidence of long period terms with large amplitude in the mean longitudes: 12 000″ in Saturn longitude, 20 000″ in that of Uranus.

Long-period modulated superstructures in Pd-12.5 at.%Ce and Pd-15 at.%Ce alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 164-165
Author(s):  
Noriyuki Kuwano ◽  
Masaru Itakura ◽  
Kensuke Oki
Keyword(s):  
Electron Microscopy ◽  
Mean Value ◽  
Heavy Elements ◽  
Arc Furnace ◽  
X Ray ◽  
Long Period ◽  
Ultra High Purity ◽  
Heat Treated ◽  
Atomic Scattering ◽  
The Mean

Pd-Ce alloys exhibit various anomalies in physical properties due to mixed valences of Ce, and the anomalies are thought to be strongly related with the crystal structures. Since Pd and Ce are both heavy elements, relative magnitudes of (fcc-fpd) are so small compared with <f> that superlattice reflections, even if any, sometimes cannot be detected in conventional x-ray powder patterns, where fee and fpd are atomic scattering factors of Ce and Pd, and <f> the mean value in the crystal. However, superlattices in Pd-Ce alloys can be analyzed by electron microscopy, thanks to the high detectability of electron diffraction. In this work, we investigated modulated superstructures in alloys with 12.5 and 15.0 at.%Ce.Ingots of Pd-Ce alloys were prepared in an arc furnace under atmosphere of ultra high purity argon. The disc specimens cut out from the ingots were heat-treated in vacuum and electrothinned to electron transparency by a jet method.

scholarly journals Exploring the robustness of Keplerian signals to the removal of active and telluric features

2020 ◽  
Vol 500 (1) ◽  
pp. 548-557
Author(s):  
M Lisogorskyi ◽  
H R A Jones ◽  
F Feng ◽  
R P Butler ◽  
S Vogt
Keyword(s):  
Monte Carlo ◽  
High Resolution ◽  
Monte Carlo Simulations ◽  
Large Amplitude ◽  
Absorption Cell ◽  
Stellar Rotation ◽  
Long Period ◽  
Active Line ◽  
Low Amplitude ◽  
Rotation Signal

ABSTRACT We examine the influence of activity- and telluric-induced radial velocity (RV) signals on high-resolution spectra taken with an iodine absorption cell. We exclude 2-$\mathring{\rm A}$ spectral chunks containing active and telluric lines based on the well-characterized K1V star α Centauri B and illustrate the method on Epsilon Eridani – an active K2V star with a long-period, low-amplitude planetary signal. After removal of the activity- and telluric-sensitive parts of the spectrum from the RV calculation, the significance of the planetary signal is increased and the stellar rotation signal disappears. In order to assess the robustness of the procedure, we perform Monte Carlo simulations based on removing random chunks of the spectrum. Simulations confirm that the removal of lines impacted by activity and tellurics provides a method for checking the robustness of a given Keplerian signal. We also test the approach on HD 40979, which is an active F8V star with a large-amplitude planetary signal. Our Monte Carlo simulations reveal that the significance of the Keplerian signal in the F star is much more sensitive to wavelength. Unlike the K star, the removal of active lines from the F star greatly reduces the RV precision. In this case, our removal of a K star active line from an F star does not a provide a simple useful diagnostic because it has far less RV information and heavily relies on the strong active lines.

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