Orbital Motion Research Articles

Dynamics and capability of along orbit debris self-collecting with rotating constant magnets

Orbital debris with equivalent size of 1∼10 cm is hard to be detected and its number is huge, which is the most harmful to on-orbit spacecraft while there is still no effective removal means. Inspired by vacuum cleaner, applying the conductive property of orbital debris and the physical law that variable magnetic flux would induce eddy current on the surface of a conductive material, this paper put forward a novel approach of ‘along orbit debris self-collecting with rotating constant magnetic dipole array’. Based on dynamic modeling, self-collecting feasibility and control capability analysis, evolution law of relative orbital motion with induced magnetic force derivation, and on-orbit debris magnetic collecting numerical simulation, this novel approach has been verified and the corresponding requirements of magnetic collector are put forward. In addition, this paper also gave how to use several key parameters to adjust the region of along orbit debris self-collecting, and pointed out several problems needing to be researched in-depth for actual application.

Impact of seasonal variations on low and mid-latitudes ionospheric electric field satellite measurements

Ionospheric electric field measurements are provided by the Langmuir probe onboard the DEMETER satellite. Because there are four probes, various combinations of them form double probes that yield measurements. However, only the combination of probe antennas (E3) and (E4) enables possible measurements of the ambient (natural) ionospheric electric field. For other combinations, the measured field is almost identical to the induced field resulting from satellite motion. On the basis of this, we conducted comparative studies of the measured and induced electric fields during different seasons (summer, autumn, winter, and spring) to determine the best period for ionospheric electric field measurements. Multiple orbits were selected during different periods when the satellite was exposed to solar radiation. In all seasons except winter, the profiles of the induced and measured ionospheric electric field are similar. This masking effect leads to an underestimation of the ambient electric field. However, the highest accuracy for the ambient field can be obtained during winter, when solar flux radiation is weaker than that during other seasons. The probe position and orbital altitude also play important roles in the behavior of the induced electric field.

Probing the eccentricity in protostellar discs: Modelling kinematics and morphologies

Context. Protostellar discs are mostly modelled as circular structures of gas and dust orbiting a protostar. However, a number of physical mechanisms, for example, the presence of a (sub)stellar companion or initial axial asymmetry, can cause the gas and dust orbital motion to become eccentric. Theoretical studies have revealed that, when present, disc eccentricity is expected to occur with predictable profiles that can be long-lasting and potentially observable in protostellar systems. Aims. We construct an analytical model predicting the typical features of the kinematics and morphology of eccentric protostellar discs, with the final goal of characterising the observational appearance of eccentricity in discs. Methods. We validate the model using a numerical simulation of a circumbinary disc (where the binary makes the disc eccentric). We finally post-process the simulation with Monte Carlo radiative transfer to study how eccentric features would appear through the ‘eyes’ of ALMA. Results. Besides the motion of the material on eccentric Keplerian orbits in the disc orbital plane, the most characteristic eccentric feature emerging from the analytical model is strong vertical motion with a typical anti-symmetric pattern (with respect to the disc line of pericentres). A circumbinary disc with a ≈ 40 au eccentric cavity (ecav = 0.2), carved by an abin = 15 au binary, placed at a distance d = 130 pc, is expected to host in its upper emission surface vertical oscillations up to vz ~ 400 m s−1 close to the cavity edge, that is to say, well within ALMA spectral and spatial resolution capabilities. A residual spiral pattern in the vertical velocity Δvz ~ 150 m s−1 of the simulation cannot be captured by the theoretical model, we speculate it to be possibly linked to the presence of a companion in the system.

Cepheids with giant companions

Context. Binary Cepheids with giant companions are crucial for studying the physical properties of Cepheid variables, in particular providing the best means to measure their masses. Systems composed of two Cepheids are even more important, but to date, only one such system has been identified, in the Large Magellanic Cloud (LMC). Aims. Our current aim is to increase the number of these systems known tenfold and to provide their basic characteristics. The final goal is to obtain the physical properties of the component Cepheids, including their masses and radii, and to learn about their evolution in the multiple systems, also revealing their origin. Methods. We started a spectroscopic monitoring campaign of nine unresolved pairs of Cepheids from the OGLE catalog to check if they are gravitationally bound. Two of these so-called double Cepheids are located in the LMC, five are in the Small Magellanic Cloud (SMC), and two are in the Milky Way (MW). Results. We report a spectroscopic detection of the binarity of all nine of these double Cepheids with orbital periods ranging from 2 to 18 years. This increases the number of known binary double (BIND) Cepheids from 1 to 10 and triples the number of all confirmed double-lined binary (SB2) Cepheids. For five BIND Cepheids, the disentangled pulsational light curves of the components show anti-correlated phase shifts due to orbital motion. We show the first empirical evidence that typical period–luminosity relations (PLRs) are rather binary Cepheid PLRs, as they include light of the companion. Conclusions. The statistics of pulsation period ratios of BIND Cepheids do not agree with those expected for pairs of Cepheids of the same age. These ratios together with the determined mass ratios far from unity suggest a merger origin of at least one component for about half of the systems. The SMC and MW objects are the first found in SB2 systems composed of giants in their host galaxies. The Milky Way BIND Cepheids are also the closest such systems, being located at about 11 and 26 kpc.

VELOcities of CEpheids (VELOCE)

We present the first data release of VELOcities of CEpheids (VELOCE), dedicated to measuring the high-precision radial velocities (RVs) of Galactic classical Cepheids (henceforth, Cepheids). The first data release (VELOCE DR1) comprises 18 225 RV measurements of 258 bona fide classical Cepheids on both hemispheres collected mainly between 2010 and 2022, along with 1161 observations of 164 stars, most of which had previously been misclassified as Cepheids. The median per-observation RV uncertainty for Cepheids is 0.037 km s−1 and reaches 2 m s−1 for the brightest stars observed with Coralie. Non-variable standard stars were used to characterize RV zero-point stability and to provide a base for future cross-calibrations. We determined zero-point differences between VELOCE and 31 literature data sets using template fitting, which we also used to investigate linear period changes of 146 Cepheids. In total, 76 spectroscopic binary Cepheids and 14 candidate binary Cepheids were identified using VELOCE data alone, which are investigated in detail in a companion Paper (VELOCE II). VELOCE DR1 provides a number of new insights into the pulsational variability of Cepheids, most importantly: a) the most detailed description of the Hertzsprung progression based on RVs to date; b) the identification of double-peaked bumps in the pulsation curve; and c) clear evidence that virtually all Cepheids feature spectroscopic variability signals that lead to modulated RV variability at the level of tens to hundreds of m s−1 and that cannot be satisfactorily modeled using single-periodic Fourier series. We identified 36 stars exhibiting such modulated variability, of which 4 also exhibit orbital motion. Linear radius variations depend strongly on pulsation period and a steep increase in slope of the ΔR/p vs. log P-relation is found near 10 days. This effect, combined with significant RV amplitude differences at fixed period, challenges the existence of a tight relation between Baade-Wesselink projection factors and pulsation periods. We investigated the accuracy of RV time series measurements, υγ, and RV amplitudes published by Gaia’s third data release (Gaia DR3) and determined an offset of 0.65 ± 0.11 km s−1 relative to VELOCE. Whenever possible, we recommend adopting a single set of template correlation parameters for distinct classes of large-amplitude variable stars to avoid systematic offsets in υγ among stars belonging to the same class. The peak-to-peak amplitudes of Gaia RVs exhibit significant (16%) dispersion. Potential differences of RV amplitudes require further inspection, notably in the context of projection factor calibration.

Precise Mass, Orbital Motion, and Stellar Properties of the M-dwarf Binary LP 349−25AB

LP 349−25 is a well-studied close stellar binary system comprised of two late M dwarf stars, where both stars are close to the limit between star and brown dwarf. This system was previously identified as a source of gigahertz radio emission. We observed LP 349−25AB over 11 epochs in 2020–2022, detecting both components in this nearby binary system using the Very Long Baseline Array (VLBA). We fit simultaneously the VLBA absolute astrometric positions together with existing relative astrometric observations derived from optical/infrared observations with a set of algorithms that use nonlinear least-squares, genetic algorithm, and Markov Chain Monte Carlo methods to determine the orbital parameters of the two components. We find the masses of the primary and secondary components to be 0.08188 ± 0.00061 M ⊙ and 0.06411 ± 0.00049 M ⊙, respectively, representing one of the most precise mass estimates of any ultracool dwarf (UCD) to date. The primary is a UCD of 85.71 ± 0.64 M Jup, while the secondary has a mass consistent with being a brown dwarf of 67.11 ± 0.51 M Jup. This is one of the very few direct detections of a brown dwarf with VLBA observations. We also find a distance to the binary system of 14.122 ± 0.057 pc. Using stellar evolutionary models, we find the model-derived stellar parameters of both stars. In particular, we obtain a model-derived age of 262 Myr for the system, which indicates that LP 349−25AB is composed of two pre–main-sequence stars. In addition, we find that the secondary star is significantly less evolved than the primary star.

Sgr A*, M87* and OJ287 as Extreme Central Binaries of Supermassive Black Holes- -Confirmation and extension of decameter radio wave observation results through VLBI data comparison

Based on a deduced 2153±5 s orbiting period of the extreme central binary of the supermassive black hole (ECB-SMBH), at Sgr A*,that is confirmed being based on the decameter radio wave pulse observations compared with 1.3 mm wavelength very long baseline interferometry (VLBI) data from the event horizon telescope (EHT-Data), we depict moving images of Sgr A* from EHT-Data, confirming further the existence of ECB-SMBH. The results show that radio wave images of the two members of ECB-SMBH are radio-bright objects due to emissions caused by source agents accreting from a thick disk toward ECB-SMBH. We extend the confirmed concept of ECB-SMBH to the interpretation of M87*, based on two images released by Miyoshi et al. in 2022 using EHT-Data and by Lu et al. in 2023 using the 3.5 mm wavelength data from the Global Millimeter VLBI Arrays as core telescopes. For the double brilliant spots displayed in two different images of M87*, we consider them the same two objects that shift locations during the 369-day interval of the observation periods of the two images. It is concluded that M87* A and M87* B (temporarily named) are orbiting with a period of 168.8 d at velocities of (6.28±0.54)% and (16.7±1.5)% of the speed of light, respectively. For the existence of ECB-SMBH, the condition of “no gravitational wave radiation” from the SMBH is essential. Regarding the secondary BH of blazar OJ 287, we find around 58~60-year period oscillation of the orbiting period, which is recently in a shortening phase. By applying the concept of ECB-SMBH to the OJ287 primary BH, whose orbiting period is approximately 1.6 yr, we interpret that the orbital period variation of the secondary BH is not due to the radiation of gravitational waves but is caused by the orbital motions of ECB-SMBH

Adapting the process conditions and performance in planetary roller melt granulation (PRMG) via the planetary spindle number

Continuous melt granulation with a planetary roller granulator is a new continuous method based on the orbital motion of planetary spindles driven by a rotating central spindle in a surrounding roller cylinder. Here the configuration of the processing section is a central design aspect, which defines the processing conditions and process performance. In this study, the number of applied planetary spindles as part of the module configuration was varied.As a result, a correlation between direct process parameters, processing conditions and process performance was identified. Hereby, the data normalization relativized successfully the effect of the module configuration. The results are also suitable to identify a module configuration and process settings in the context of a process optimization towards energy input efficiency or energy consumption. Finally, the accounting of the particle modification in terms of size was suitable to link the effect of the investigated parameters to the granulation regime.

Simultaneous geometric calibration and orbit-attitude determination of Hayabusa2’s deployable camera (DCAM3)

Hayabusa2’s deployable camera (DCAM3) was deployed from the spacecraft near the asteroid Ryugu to successfully monitor the artificial cratering experiment. Scientific analyses of the observed impact ejecta require accurate determination of the camera’s position and orientation. However, in contrast to conventional spacecraft operations, DCAM3 lacked the capability to acquire orbit tracking data and attitude sensor data due to its limited onboard resources. Even though the optical images obtained by DCAM3 itself are the only source of geometric information, they were subject to significant distortion caused by the rolling shutter effect and lens distortion. This research is, therefore, designed to simultaneously estimate the image distortion and the orbital and attitude motions of DCAM3, relying solely on its image data. The relative geometry between the camera and the asteroid is reconstructed from the geographic information of the observed feature points by incorporating distortion effects. The proposed method involves segmenting the geometry reconstruction, with more than ten thousand feature measurements, into smaller-scale least-squares problems. The image-based estimation yields consistent distortion, orbit, and attitude solutions with pixel-scale accuracy. The analysis results indicate that DCAM3 experienced significant nutation of its optical axis during ballistic flight in a semi-elliptical orbit. In addition, these motions were sensitive enough to simultaneously determine system parameters, such as the gravitational parameter of Ryugu and the inertia moment of DCAM3. This paper illustrates the potential of deployable camera systems as a promising option for enhancing the scientific and engineering aspects of asteroid exploration.

Detecting New Visual Binaries in Gaia DR3 with Gaia and Two Micron All Sky Survey (2MASS) Photometry. II. Speckle Observations of 16 Low-separation Systems

Here we present speckle observations of 16 low-separation (s < 30 au) high-probability candidate binaries from the catalog by Medan et al., where secondaries typically lack astrometric solutions in Gaia. From these speckle observations, we find a second component is always detected within the field of view. To determine if the detection is consistent with a physical companion or a chance alignment with a background source, we utilize a statistic from Tokovinin & Kiyaeva that compares the apparent motion of the systems to the expected orbital motion ( μ′ ). Using simulated binary orbits, we construct likelihood distributions of μ′ assuming various total errors on the measurements. With the hypothesis that the system is a true binary, we show that large measurement errors can result in μ′ values higher than expected for bound systems. Using simulated chance alignments, we also create similar likelihoods to test this alternative hypothesis. By combining likelihoods of both true binaries and chance alignments, we find that 15 of the 16 candidates are physical systems regardless of the level of measurement error. Our findings also accommodate all 16 as physical systems if the average, relative measurement error on the binary separations and position angles is ∼4.3%, which is consistent with our knowledge of the Gaia and Gemini speckle pipelines. Importantly, beyond assessing the likelihood of a true binary versus chance alignment, this quantitative assessment of the true average measurement error will allow more robust error estimates of mass determinations from short separation binaries with Gaia and/or Gemini speckle data.

Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres.

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.

Eccentric binaries: Periastron events and tidal heating

Context. Binary stars cannot be fully understood without assessing the interaction effects between the two components and the impact of these effects on observational diagnostics. Periastron brightening events, also known as the heartbeat phenomenon, are a clear manifestation of this type of interaction. Aims. We aim to explore the role of tidal shear energy dissipation in stars undergoing periastron brightening events. Methods. We performed a computation from first principles that uses a quasi-hydrodynamic Lagrangian scheme to simultaneously solve the orbital motion and the equations of motion of a 3D grid of volume elements covering the inner, rigidly rotating “core” of a tidally perturbed star. The equations of motion include the gravitational acceleration of both stars, the centrifugal, Coriolis, gas pressure accelerations, and viscous coupling between volume elements. The method is illustrated for a grid of model binary systems with a 10 M⊙ primary that is perturbed by a 6.97 M⊙ companion in eccentric orbits (e = 0 − 0.7). The model is then applied to the heartbeat star MACHO 80.7443.1718. Results. We find an increase by factors ∼10−6–10−3 in tidal shear energy dissipation at periastron, consistent with the majority of observed heartbeat stars. The magnitude of the periastron effect correlates with the degree of departure from synchronicity: stars rotating much faster or much slower than the synchronous rate at periastron present the strongest effect. We confirm that for eccentricities ≤0.3, pseudo-synchronization occurs for 0.8 &lt; ω/Ωave &lt; 1, where Ωave is the average orbital angular velocity. The minimum energy rotation rate (pseudo-synchronism) for e = 0.5 and 0.7 occurs for 1.0 &lt; ω/Ωave &lt; 1.15. The tidal shear energy dissipation model reproduces from first principles the ∼23% maximum brightness enhancement at periastron of MACHO 80.7443.1718. Conclusions. Our results suggest that the magnitude and shape of the heartbeat signal may serve as diagnostics for the internal stellar rotation and turbulent viscosity values.

Transitory Tidal Heating and Its Impact on Cluster Isochrones

The kinetic energy in tidal flows, when converted into heat, can affect the internal structure of a star and shift its location on a color–magnitude diagram from that of standard models. In this paper we explore the impact of injecting heat into stars with masses near the main-sequence turnoff mass (1.26 M ⊙) of the open cluster M67. The heating rate is obtained from the tidal shear energy dissipation rate, which is calculated from first principles by simultaneously solving the equations that describe orbital motion and the response of a star’s layers to the gravitational, Coriolis, centrifugal, gas pressure, and viscous forces. The stellar structure models are computed with MESA. We focus on the effects of injecting heat in pulses lasting 0.01 Gyr, a time frame consistent with the synchronization timescale in binary systems. We find that the location of the tidally perturbed stars in the M67 color–magnitude diagram is shifted to significantly higher luminosities and effective temperatures than predicted by the standard model isochrone and includes locations corresponding to some of the blue straggler stars. Because tidal heating takes energy from the orbit, causing it to shrink, blue straggler stars could be merger or mass transfer progenitors, as well as products of these processes.

Numerical model of Phobos’ motion incorporating the effects of free rotation

Context. High-precision ephemerides are not only useful in supporting space missions, but also in investigating the physical nature of celestial bodies. This paper reports an update to the orbit and rotation model of the Martian moon Phobos. In contrast to earlier numerical models, this paper details a dynamical model that fully considers the rotation of Phobos. Here, Phobos’ rotation is first described by Euler’s rotational equations and integrated simultaneously with the orbital motion equations. We discuss this dynamical model, along with the differences with respect to the model now in use. Aims. This work is aimed at updating the physical model embedded in the ephemerides of Martian moons, considering improvements offered by exploiting high-precision observations expected from future missions (e.g., Japanese Martian Moons exploration, MMX), which fully supports future studies of the Martian moons. Methods. The rotational motion of Phobos can be expressed by Euler’s rotational equations and integrated in parallel with the equations of the orbital motion of Phobos around Mars. In order to investigate the differences between the two models, we first reproduced and simulated the dynamical model that is now used in the ephemerides, but based on our own parameters. We then fit the model to the newest Phobos ephemeris published by Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE). Based on our derived variational equations, the influence of the gravity field, the Love number, k2, and the rotation behavior were studied by fitting the full model to the simulated simple model. Our revised dynamic model for Phobos was constructed as a general method that can be extended with appropriate corrections (mainly rotation) to systems other than Phobos, such as the Saturn and Jupiter systems. Results. We present the variational equation for Phobos’ rotation employing the symbolic Maple computation software. The adjustment test simulations confirm the latitude libration of Phobos, suggesting gravity field coefficients obtained using a shape model and homogeneous density hypothesis should be re-examined in the future in the context of dynamics. Furthermore, the simulations with different k2 values indicate that it is difficult to determine k2 efficiently using the current data.

Constraining stellar tidal quality factors from planet‐induced stellar spin‐up

AbstractThe dynamical evolution of tight star‐planet systems is influenced by tidal interactions between the star and the planet, as was shown recently. The rate at which spins and orbits in such a system evolve depends on the stellar and planetary tidal dissipation efficiency. Here, we present a method to constrain the modified tidal quality factor of a planet‐hosting star whose rotational evolution has been altered by its planet through angular momentum transfer from the planetary orbital motion to the rotation of the stellar convective zone. The altered rotation is estimated from an observed discrepancy of magnetic activity of the planet‐hosting star and a coeval companion star, that is, this method is applicable to star‐planet systems with wide stellar companions. We give an example of the planet‐hosting wide binary system HD189733 and find that the planet host's modified tidal quality factor is constrained to be .

A simplified analytical approach for determining eclipses of satellites occulted by a celestial body

The primary objective of this paper is to construct an analytical model for determining the total duration of eclipse events of satellites. The approach is based on the assumption that the trace formed in the orbital plane, cutting body’s shadow under the classical conical shadow model, can be described as an ellipse. This allows for the derivation of its parameters through the use of classical orbital elements and solar position in the body’s centric frame. Consequently, the problem of identifying satellite occultation events is simplified to searching for points of intersection between two ellipses: the satellite orbit and the shadow. The developed model has been demonstrated to be applicable for a wide range of inclinations, with the exception of cases where the orbital plane of the satellite's motion coincides with that of the sun's in the body-centric frame. It is shown that the shadow function constructed under the aforementioned assumptions can be simplified in the cylindrical shadow model. Several simplifications are provided in the research. The paper also considers the extension of the model to account for the non-spherical shape of celestial bodies and presents an algorithm for accounting for changes in eclipse duration caused by the heliocentric motion of celestial bodies. The model has been validated through numerous tests with a numerical algorithm and satellite real data, as well as with other analytical models.

Introduction of the Runge-Kutta method in GPS orbit computation

In all Global Navigation Satellite Systems (GNSS) applications, the determination of the satellite orbits is an important task. In this study, we present the equations given in the Interface Specification Document of GPS and the Runge-Kutta method in the computation of the position P, velocity V, and acceleration A of the GPS satellites using the broadcast ephemeris. The definition of the differential equation describing the GPS satellite's motion has enabled us to introduce the Runge-Kutta method in the GPS orbit computation; this method uses the initial conditions determined in this study from the Keplerian elements provided in the broadcast ephemeris files. The Lagrange interpolation method is used for comparison of the results, where the vectors P, V, and A are estimated using the precise ephemeris. The difference not exceeding 2.4 m was obtained in the X, Y, and Z axes during seven days on the position of the GPS satellite number 9 tested in this study. In velocity and acceleration, the difference is about a few mm/s and mm/s2, respectively.

Exploring the physical properties of the [formula omitted] Dor binary star RX Dra with photometry and asteroseismology

We model the TESS light curve of the binary system RX Dra, and also first calculate a lot of theoretical models to fit the g-mode frequencies previously detected from the TESS series of this system. The mass ratio is determined to be q=0.9026−0.0032+0.0032. We newly found that there are 16 frequencies (F1–F7, F11–F20) identified as dipole g-modes, two frequencies (F21, F22) identified as quadrupole g-modes, and another two frequencies (F23, F24) identified as g-sextupole modes, based on these model fits. The primary star is newly determined to be a γ Dor pulsator in the main-sequence stage with a rotation period of about 5.7−0.2+0.7 days, rotating slower than the orbital motion. The fundamental parameters of two components are firstly estimated as follows: M1=1.53−0.17+0.00 M ⊙, M2= 1.38−0.00+0.18 M ⊙, T1=7240−44+490 K, T2=6747−221+201 K, R1=1.8288−0.0959+0.0260 R ⊙, R2= 1.3075−0.2543+0.0450 R ⊙, L1=8.2830−0.6036+1.8015 L ⊙ and L2= 3.4145−0.1843+0.1320 L ⊙. Our results show that the secondary star is in the solar-like pulsator region of the H–R diagram, indicating that it could be a pulsating star of this type. Finally, the radius of the convective core of the primary star is estimated to be about 0.1403−0.0000+0.0206 R ⊙.

The Dynamical Equations of the Restricted Three-Body Problem with Poynting-Robertson Drag Force and Variable Masses

The restricted three-body problem (R3BP) is a formulation which defines the motion of a passively gravitating test particle having infinitesimal mass and moving in the gravitational environment of two bodies, called primaries. The R3BP is still an exciting and active research field that has been getting attention of scientists and astronomers because of its applications in dynamics of the solar and stellar systems, lunar theory, and artificial satellites. The equations of motion are usually the starting point in the investigations of the dynamical predictions of the infinitesimal mass. Therefore, in this paper, we examine the derivations of the dynamical equations of the R3BP with Poynting-Robertson (P-R) Drag force and variable masses. In this model formulation, both primaries are assumed to vary their masses under the combined Mestschersky law (CML) and they move in the frame of the GyldenMestschersky equation (GME). Further, the bigger primary is assumed to be emitting radiation force, which is a component of the radiation pressure and the P-R drag. The non- autonomous dynamical equations of the model are derived and converted into the autonomized equations with constant coefficients using the Mestschersky transformation (MT), the CML, the particular solutions of the GMP, and a transformation for the time dependent velocity of light. We observed that the P-R drag of the bigger primary depends on the mass parameter, radiation pressure, velocity of light and the mass variation constant . The derived systems of equations with variable and constant coefficients can be used to model the long-term motion of satellites and planets in binary systems.

A simultaneous measurement method for multi-directional galloping of iced bundled conductors and its application in spatial galloping behavior

The theoretical results indicate that iced bundled conductors experience spatial galloping due to wind-induced vibration, involving in-plane, out-of-plane, and torsional movements. To better understand the dynamic response of this behavior from an experimental perspective, an innovative experimental method has been proposed. The method can simultaneously measure the in-plane, out-of-plane, and torsional vibration signals of iced bundled conductors’ galloping. A testing system was established, and the method is applied in the galloping experiment of continuous iced bundled conductors, validating some theoretical results. This paper describes the construction of a wide-aperture, low-speed wind tunnel suitable for testing the galloping of continuous iced bundled conductors. A ‘cut-bury-glue’ method was proposed to create a model of continuous iced bundled conductors effectively, along with a method for connecting subconductors to improve experimental precision. The tests utilized laser displacement sensors and wireless posture sensors, considering the installation and data collection characteristics of the sensors. Through signal conversion, error correction, and other technical methods, simultaneous measurement of in-plane, out-of-plane, and torsional vibration signals was achieved. The spatial galloping behavior, changing with wind speed, exhibits limited-amplitude and synchronous characteristics. The participation of different modes shows elliptical orbital motion in single-mode galloping and ‘8’ shaped orbital motion in coupled-mode galloping. These results are consistent with previous theoretical research, offering a new approach to studying iced bundled conductors’ galloping.