Small Periodic Variations Research Articles

Phase-locked and phase-unlocked multicolor solitons in a fiber laser.

Multicolor solitons are nonlinear pulses composed of two or more solitons centered at different frequencies, propagating with the same group velocity. In the time domain, multicolor solitons consist of an envelope multiplying a more rapidly varying fringe pattern that results from the interference of these frequency components. Here, we report the observation in a fiber laser of a novel, to the best of our knowledge, type of dynamics in which different frequency components still have the same group velocity but have different propagation constants. This causes the relative phases between the constituent spectral components to change upon propagation, corresponding to the fringes moving under the envelope. This leads to small periodic energy variations that we directly measure. Our experimental results are in good agreement with realistic numerical simulations based on an iterative cavity map.

Revisiting the Cassini States of synchronous satellites with an angular momentum approach

AbstractLike our Moon, the large icy satellites of Jupiter are thought to be in a Cassini State, an equilibrium rotation state characterized by a synchronous rotation rate and a precession rate of the rotation axis equal to that of the normal to the orbit. In these equilibrium states (up to four Cassini States are possible for a solid and rigid satellite), the spin axis of the satellite, the normal to its orbit and the normal to the inertial plane remain coplanar with an obliquity that remains theoretically constant. However, as the gravitational torque exerted on the satellite shows small periodic variations, the orientation of the rotation axis will also vary with time and nutations in obliquity will appear.Here we present a dynamical model for the study of the Cassini States. This model includes the coupling between the polar motion and the spin axis precession/nutation which is neglected in the classical studies. We study the influence of the triaxiality of Ganymede on its four possible Cassini States, use a Toy model of the Moon to illustrate the nutations in obliquity obtained with the dynamical model, and investigate the influence of the presence of a subsurface ocean on the Cassini State I of Europa.

Influence of lunisolar tides on plants. Parametric resonance induced by periodic variations of gravity

Recent experiments conducted in the International Space Station highlight the apparent periodicity of leaf oscillations and other biological phenomena associated with rhythmic variations of lunisolar forces. These events are similar to those occurring on Earth but with greater effects over a shorter period of time. Among the possible disturbances, other than forced or self-existing oscillations, parametric resonances appear caused by a small periodic term; such is the case of fluids subjected to small periodic variations in gravitational forces in microscopic or mesoscopic plant channels filled with sap and air-vapor. The interface instabilities verify Mathieu’s second order differential equation resulting from a Rayleigh–Taylor stability model. These instabilities appear during the Moon’s rotation around the Earth and during the revolution of the International Space Station. They create impulses of pressure and sap movements in the network of roots, stems, and leaves. The model can explain the effects of the lunar tide on plant growth. The eccentricity of the lunar orbit around the Earth creates an important difference between the apogee and perigee of the Moon’s trajectory, and therefore, the tidal effects can depend on the distance between the Moon and the Earth.

Nonequilibrium calorimetry

We consider stationary driven systems in contact with a thermal equilibrium bath. There is a constant (Joule) heat dissipated from the steady system to the environment as long as all parameters are unchanged. As a natural generalization from equilibrium thermodynamics, the nonequilibrium heat capacity measures the excess in that dissipated heat when the temperature of the thermal bath is changed. To improve experimental accessibility we show how the heat capacity can also be obtained from the response of the instantaneous heat flux to small periodic temperature variations.

Structural periodicity in laser additive manufactured Zr-based bulk metallic glass

Additive manufacturing of bulk metallic glasses (BMGs) allows for effective bypassing of critical casting thickness constraints for glassy alloys, opening up this exciting material class to new applications. An open question is how the laser processing of such materials affects the short-range structural order, a critical mediating parameter for glass deformation. Synchrotron X-ray microdiffraction was used to understand structural heterogeneity across the build-planes of a selective laser melted Zr-based BMG. While negligible macroscopic heterogeneity in the structure was observed over a 10 mm build height for the X-ray amorphous material, small periodic variations were observed on the order of 40–80 μm. This dimensional scale was rationalized as a consequence of melt-pool solidification from laser processing, which imparts a calculated local strain variation of ±0.1%. It is anticipated that this structural insight will help to rationalize microscale deformation effects from the periodic structural variation of selective laser melting produced BMGs.

Seasonal variations of decay rate measurement data and their interpretation

Measurement data of long-lived radionuclides, for example, 85Kr, 90Sr, 108mAg, 133Ba, 152Eu, 154Eu and 226Ra, and particularly the relative residuals of fitted raw data from current measurements of ionization chambers for half-life determination show small periodic seasonal variations with amplitudes of about 0.15%. The interpretation of these fluctuations is a matter of controversy whether the observed effect is produced by some interaction with the radionuclides themselves or is an artifact of the measuring chain. At the origin of such a discussion there is the exponential decay law of radioactive substances used for data fitting, one of the fundamentals of nuclear physics. Some groups of physicists use statistical methods and analyze correlations with various parameters of the measurement data and, for example, the Earth-Sun distance, as a basis of interpretation. In this article, data measured at the Physikalisch-Technische Bundesanstalt and published earlier are the subject of a correlation analysis using the corresponding time series of data with varying measurement conditions. An overview of these measurement conditions producing instrument instabilities is given and causality relations are discussed. The resulting correlation coefficients for various series of the same radionuclide using similar measurement conditions are in the order of 0.7, which indicates a high correlation, and for series of the same radionuclide using different measurement conditions and changes of the measuring chain of the order of −0.2 or even lower, which indicates an anti-correlation. These results provide strong arguments that the observed seasonal variations are caused by the measuring chain and, in particular, by the type of measuring electronics used.

Wavelet Analysis on the Temporal Series of Precipitation in Baoji

Adopting the series data of precipitationfrom 1952 to 2013 in Baoji city, shaanxi province, China, using Morlet wavelet function, the seasonal variation of precipitation and the time series of the interannual variation of precipitation in Baoji nearly 60aare analyzed by wavelet analysis, reveals the multiple time scales change complex structure of Baoji precipitation, forecasts the precipitation change trend of four seasons in Baoji city. The results show that the seasonal and annual precipitationin Baoji has the characteristics of multi time scale, different scales show the different cycles, large scale periodic variations also include small scale periodic variations. As a whole, the performance for small scale changed seriously, no clear rules of special features, and there is obvious regularity of large scale. The time-frequency localization characteristic of wavelet analysis can show the fine structure of precipitation time series, and provide a new method for the analysis of the key water saving problems, such as multi time scale variation characteristics and short-term climate prediction.

Boundary Layer for a Non-Newtonian Flow Over a Rough Surface

We analyze how small irregularities of a solid wall affect the steady flow of a non-Newtonian fluid. We consider the generalized Stokes system for power-law type fluids, with no-slip boundary conditions. Irregularities are modeled by small periodic variations of the boundary surface, described by a small parameter. We derive an effective boundary condition---a wall law---on a smoothed boundary, with the smallest possible approximation error (in terms of the parameter). The keypoint of the mathematical analysis is the study of the boundary layer generated by the irregularities. We stress that our results apply both to shear thickening and shear thinning fluids.

Using Galactic Cepheids to verify Gaia parallaxes

Context. The Gaia satellite will measure highly accurate absolute parallaxes of hundreds of millions of stars by comparing the parallactic displacements in the two fields of view of the optical instrument. The requirements on the stability of the 'basic angle' between the two fields are correspondingly strict, and possible variations (on the microarcsec level) are therefore monitored by an on-board metrology system. Nevertheless, since even very small periodic variations of the basic angle might cause a global offset of the measured parallaxes, it is important to find independent verification methods. Aims. We investigate the potential use of Galactic Cepheids as standard candles for verifying the Gaia parallax zero point. Methods. We simulate the complete population of Galactic Cepheids and their observations by Gaia. Using the simulated data, simultaneous fits are made of the parameters of the period-luminosity relation and a global parallax zero point. Results. The total number of Galactic Cepheids is estimated at about 20 000, of which nearly half could be observed by Gaia. In the most favourable circumstances, including negligible intrinsic scatter and extinction errors, the determined parallax zero point has an uncertainty of 0.2 microarcsec. With more realistic assumptions the uncertainty is several times larger, and the result is very sensitive to errors in the applied extinction corrections. Conclusions. The use of Galactic Cepheids alone will not be sufficient to determine a possible parallax zero-point error to the full potential systematic accuracy of Gaia. The global verification of Gaia parallaxes will most likely depend on a combination of many different methods, including this one.

A scheme for lunar inner core detection

Very precise measurements of the lunar gravity field could detect a solid inner core. For synchronous rotation, the equator planes of the inner core and mantle should be tilted with respect to the ecliptic plane and precess along that plane with the same 18.6 yr period. Generally, two different tilts would cause a static inner core gravity field to appear as small periodic (27.212 d) variations in the mantle‐referenced C21 and S21 coefficients. Tidal variations also contribute to the C21 variation so a much improved k2 Love number would be required. Model computations suggest that the inner core signature is likely to be very small requiring sensitive gravity measurements. In principle, a signature analogous to the Moon's should be present for other synchronous satellites with interior liquid layers and also Mercury.

Motion of an elliptical vortex under rotating strain: conditions for asymmetric merging

The analysis of the motion of a uniform vortex ( patch) of elliptical shape under a rotating strain field is employed to investigate the conditions leading to merging for two co-rotating elliptical patches of equal vorticity but different circulation. The motion of the patch having smaller circulation under the rotating strain field induced by the other vortex is analyzed, by considering both the intensity and the rotation rate of the strain constant. Under this assumption, we may adopt a first integral of motion which has been already used to discuss the different kinds of motion experienced by the elliptical patch. In the present paper, the dependence of the motion on the initial conditions and on the strain parameters is analyzed in further detail, to provide an overall picture of the elliptical patch dynamics. These results are employed in the analysis of the merging conditions. To this aim, the strain parameters, written in terms of the circulation of the larger vortex and of the distance between the two vortices, are kept frozen at their initial values. For a given circulation of the larger vortex and for a given vorticity and initial configuration of the smaller one, it is possible to find a particular distance between the vortices – transitional distance, d t – to which a significant change in the dynamics of the smaller elliptical patch is associated. It is always slightly lower than the critical distance and the difference between the two values decreases for vanishing size of the smaller vortex. The analysis of the first integral shows that, for an initial distance larger than d t, the motion of the vortex leads to small periodic variations of its second-order moment. On the contrary, when the distance is smaller than d t, the motion implies a large growth of its second-order moment, that is able to force the merging for frozen strain parameters.

Dynamics of Orbits Close to Asteroid 4179 Toutatis

We use a radar-derived physical model of 4179 Toutatis (975870) to investigate close-orbit dynamics around that irregularly shaped, non-principal-axis rotator. The orbital dynamics about this body are markedly different than the dynamics about uniformly rotating asteroids. The results of this paper are generally applicable to orbit dynamics about bodies in a non-principal-axis rotation state. The radar results support the hypothesis that Toutatis has a homogeneous density distribution, and we assume a density of 2.5 g/cc. The asteroid's gravity field is computed using a truncated harmonic expansion when outside of its circumscribing sphere and a closed-form expression for the potential field of an arbitrary polyhedron when inside that sphere. The complete equations of motion are time-periodic in the Toutatis-fixed frame due to the complex rotation of the asteroid. The system is Hamiltonian and has all the characteristics of such a system, including conservation of phase volume, but there is no Jacobi constant of the motion and zero velocity surfaces cannot be used to analyze the system's behavior. We also examine some of the close-orbit dynamics with the Lagrange planetary form of the equations of motion. Families of quasi-periodic “frozen orbits” that show minimal variations in orbital elements are found to exist very close to the asteroid; some of them are stable and hence can hold natural or artificial satellites. A retrograde family of frozen orbits is especially robust and persists down to semi-major axes of about 2.5 km, comparable to half of Toutatis' longest dimension. We identify families of periodic orbits, which repeat in the Toutatis-fixed frame. Due to the time-periodic nature of the equations of motion, all periodic orbits about Toutatis in its body-fixed frame must be commensurate with the 5.42-day period associated with those equations. Exact calculations of both stable and unstable periodic orbits are made. The sum of surface forces acting on a particle on Toutatis is time-varying, so particles on and in the asteroid are being continually shaken with a period of 5.42 days, perhaps enhancing the uniformity of the regolith distribution. A global map of the gravitational slope reveals that it is surprisingly shallow for such an elongated, irregularly shaped object, averaging 16° globally and less than 35° over 96% of the surface. A global map of tangential accelerations shows no values larger than 0.5 mm/s2, an average value of 0.2 mm/s2, and less than 0.25 mm/s2over 70% of the surface. A global map of the escape speed for launch normal to the surface shows that quantity to be between 1.2 and 1.8 m/s over most of the surface. Each of these mapped quantities has small periodic variations. We have found trajectories that leave the surface, persist in the region of phase space around a frozen orbit, and then impact the surface after a flight time of more than 100 days. Return orbit durations of years seem possible. Whereas a uniformly rotating asteroid preferentially accumulates non-escaping ejecta on its leading sides, Toutatis accumulates ejecta uniformly over its surface. We render a variety of close orbits in inertial and body-fixed frames.

A FIBER-FED ECHELLE SPECTROGRAPH FOR THE HALE 5-M TELESCOPE

ABSTRACT We describe a new fiber-fed echelle spectrograph which is now in operation on the Hale 5-meter telescope at Palomar Observatory. The instrument is optimized for high spectral stability, necessary for asteroseismology measurements of small periodic variations in stellar radial velocities. It features a 7-element all-spherical 610-mm focal length f/3.0 lens system and prism cross-disperser in a compact double-pass quasi-Littrow configuration. Light enters the system at prime focus, and is channeled to the spectrograph by a fiber optic cable. The instrument rests on a fixed-orientation optical bench inside the telescope's East Arm. It can be operated in a low-resolution mode, with resolution R=20,000 and overal efficiency (including atmospheric seeing) e~5 percent at 550 nm, or a high-resolution mode, with R=40,000 and e~1.5 percent at 550 nm; both modes have Rpixel=100,000. With large-format CCD detector (2048X2048 with 27-micron pixels), the entire visible spectrum from 400-1000 nm can be recorded in two exposures, with no gaps. We have also incorporated a novel imaging system to produce a circularly symmetric guide image of the input fiber tip. Using a fiber-optic double scrambler, the instrument produces radial-velocity measurements which are stable at the ~1 m/sec level over short (<~ 30 minute) time periods.

New acoustic detection technique for a magnetocaloric effect

We present a new photoacoustic-like detection technique which allows for the rapid detection of small periodic temperature variations induced by small periodic adiabatic magnetic field variations imposed on a magnetic material. Results are presented for gadolinium near the Curie point. From the results for the periodic temperature variations of the sample information can be obtained about the magnetic equation of state.

Rotation and Shape of High Altitude Reflectors Controlled from the Ground

Passive, inverted-dish shaped communications mirrors made of a thin wire mesh may be maintained in a stationary position above most of the atmosphere by the pressure of reflected radiation beamed at the mirror from the ground. We show that by beaming at the mirror an appropriate mixture of linearly and circularly polarized radiation, and by inducing small periodic variations of the plane of polarization, one can monitor and sufficiently control wire orientation in the mirror, which allows one to reduce mirror weight and/or set the mirror in rotation. Mirror rotation in turn can modify mirror shape, open up flaps, and allow synchronized messages to be sent to prechosen locations.

Stabilized flow in capillary-type closed viscometer

Small periodic variations in a main gas flow through a capillary are reduced from 1% to 0.1% in peak-to-peak width by a feedback circuit between a pressure gauge and a motor of a pump. The variations in flow result from nonlinear transmission of a worm mechanism in a capillary-type closed viscometer, Dynamical properties of the system are discussed by the Laplace transform method, and it has been shown that low pressure, a long guide tube, and high gain of the amplifier are responsible for the flow instability.

Acoustic Wave Motion along a Periodic Surface

An analysis is made of a surface wave traveling axially along the boundary of au infinitely long cylindrical surface that has small periodic variations superimposed upon it. By the application of Floquet's theorem, the acoustic pressure is expressed in terms of an infinite series. For waveguides that have acoustic reactances that are periodic with period L in the axial direction, an infinite system of equations is obtained. Unknown amplitude coefficients are determined by a method of infinite determinants. After the amplitude coefficients are known in terms of one coefficient, the phase variation for the surface wave along the cylindrical axis is calculated by Φ (z) = arctan (ImP/ReP). The general method is illustrated by considering a waveguide with a variable reactance given by χ = χ0 + x cos (2πz/L), where x≪χ0 and z is the axial coordinate. The effects of small changes in the parameters x, χ0, and L are analyzed for an arbitrary surface wave. It is shown that for a surface wave with wave number given by h = 2π/;λg, as hL → π with hL &amp;lt; π, the phase variation is quite rapid in the vicinity of z = nL, where n is a positive integer and is very small over the remainder of the waveguide.

Phase Variation of Acoustic Surface Waves on a Periodic Surface

An analysis is made of the phase variation for a surface wave traveling along a periodic surface. For an infinitely long cylindrical suface that has small periodic variations superimposed upon it, the acoustic reactance is periodic. By application of Floquet's theorem, wavemotion at the surface is described by an infinite series. A waveguide that has a constant reactance with a small periodically varying reactance superimposed upon it is used to illustrate the analysis. Infinite determinants are used to evaluate unknown amplitude coefficients that appear in the series solution. The phase variation for a surface wave with wavenumber h = 2π/λg, which is traveling along the surface with period L, is discussed. It is shown that, as hL → π−, the phase variation is very great in the vicinity of integral values of L along the cylindrical axis and it is very small along the remainder of waveguide.

The perturbation of satellite orbits by extra-terrestrial gravitation

Under the influence of the sun and moon some of the elements of a satellite orbit undergo small secular and periodic variations. These changes are expressible in terms of the Lagrangian Planetary Equations and are integrable, under certain conditions, to give the mean rates of change of the elements of the orbit over one revolution of the satellite. Subsequent integration gives any secular motions that may exist. Numerical examples show that in general the lunar perturbation is more than twice as large as that of the solar perturbation. However, it is found that the cumulative effect of the periodic variations caused by the sun can often be larger than that caused by the moon because the period of revolution of the earth about the sun is much greater than that of the moon about the earth.

On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena

1. Obvious and hidden periodicities. A variable quantity may show periodic changes which become obvious as soon as a sufficient record has been obtained; such are the semi‐diurnal changes of the tides, or the eleven years recurrence of sunspot maxima. We may call these obvious periodicities. Most often, however, small periodic variations are hidden behind irregular fluctuations, and their investigation then becomes a matter of considerable difficulty.