Laplace Plane Research Articles

Orbital instability of close-in exomoons in non-coplanar systems

This work shows the dynamical instability that can happen to close-in satellites when planet oblateness is not accounted for in non-coplanar multiplanet systems. Simulations include two secularly interacting Jupiter-mass planets mutually inclined by 10 degrees, with the host planet either oblate or spherical. With a spherical host planet, moons within a critical planetocentric distance experience high inclinations and in some cases high eccentricities, while more distant moons orbit stably with low inclinations and eccentricities, as expected. These counter-intuitive dynamical phenomena disappear with an oblate host planet, in which case the moons' Laplace plane transitions from the host planet's equatorial plane to the host planet's precessing orbital plane as their semi-major axes increase, and all moons are dynamically stable with very mild changes in orbits. Direct perturbation from the perturbing planet has been investigated and ruled out as an explanation for the behavior of the innermost satellites, therefore leaving the central star's perturbation as the cause. Instability occurs while the nodal precession of the satellite and the central star (as seen from the host planet's frame) approaches the 1:1 secular resonance. In non-coplanar systems, around a non-oblate planet, the nodal precession of the moon becomes slow and comparable to that of the planet, giving rise to resonant configurations. The above effect needs to be taken into account in setting up numerical simulations.

Locating Large Solar Power Satellites in the Geosynchronous Laplace Plane

Designs for geostationary solar power satellites are extremely large in scale, more than an order of magnitude larger than the International Space Station. A detailed study of solar power satellites’ orbit dynamics is performed, obtaining a comprehensive understanding of the effect of perturbations on orbits of large solar power satellite structures over a time frame commensurate with proposed solar power satellites’ lifetimes (30–40 years). Analytical equations derived by the process of averaging of the solar power satellites’ equations of motion are used to determine the long-term orbital behavior. Previous solar power satellite studies have simply assumed geostationary Earth orbit, then designed control systems for maintaining it thus. It is found that an alternative solar power satellite orbital location, known as the geosynchronous Laplace plane, is superior to geostationary in many aspects. A solar power satellite in the geosynchronous Laplace plane requires virtually no fuel to maintain its orbit, minimizes the risk of debris creation at geosynchronous altitude, and is extremely operationally robust (i.e., loss of control is inconsequential). The geosynchronous Laplace plane solar power satellite saves on order of per year in fuel compared with a geostationary solar power satellite for near-equivalent power delivery. It is also found that the microwave beam orbit perturbation, which is unique to the solar power satellites, contributes toward stabilization of the longitude of the solar power satellites.

Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5: 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravitational forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simultaneous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6″/yr and the period TJ = TS ≈ 50687 yr. These results are close to those obtained in the general theory (25.93″/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β1 ≈ 0.8403504° (Saturn) and β2 ≈ 0.3409296° (Jupiter) around the point which is located at an angular distance of 1.647607° from the ecliptic pole.

EXCITATION OF THE ORBITAL INCLINATION OF IAPETUS DURING PLANETARY ENCOUNTERS

Saturn's moon Iapetus has an orbit in a transition region where the Laplace surface is bending from the equator to the orbital plane of Saturn. The orbital inclination of Iapetus to the local Laplace plane is ~8 deg, which is unexpected, because the inclination should be ~0 if Iapetus formed from a circumplanetary disk on the Laplace surface. It thus appears that some process has pumped up Iapetus's inclination while leaving its eccentricity near zero (e=0.03 at present). Here we examined the possibility that Iapetus's inclination was excited during the early solar system instability when encounters between Saturn and ice giants occurred. We found that the dynamical effects of planetary encounters on Iapetus's orbit sensitively depend on the distance of the few closest encounters. In four out of ten instability cases studied here, the orbital perturbations were too large to be plausible. In one case, Iapetus's orbit was practically unneffected. In the remaining five cases, the perturbations of Iapetus's inclination were adequate to explain its present value. In three of these cases, however, Iapetus's eccentricity was excited to 0.1-0.25, and it is not clear whether it could have been damped to its present value (0.03) by some subsequent process (e.g., tides and dynamical friction from captured irregular satellites do not seem to be strong enough). Our results therefore imply that only 2 out of 10 instability cases (~20%) can excite Iapetus's inclination to its present value (~30% of trials lead to >5 deg) while leaving its orbital eccentricity low.

LAPLACE PLANE MODIFICATIONS ARISING FROM SOLAR RADIATION PRESSURE

The dynamical effects of solar radiation pressure (SRP) in the solar system have been rigorously studied since the early 1900s. This non-gravitational perturbation plays a significant role in the evolution of dust particles in circumplanetary orbits, as well as in the orbital motion about asteroids and comets. For gravitationally dominated orbits, SRP is negligible and the resulting motion is largely governed by the oblateness of the primary and the attraction of the Sun. The interplay between these gravitational perturbations gives rise to three mutually perpendicular planes of equilibrium for circular satellite orbits. The classical Laplace plane lies between the equatorial and orbital planes of the primary, and is the mean reference plane about whose axis the pole of a satellite's orbit precesses. From a previously derived solution for the secular motion of an orbiter about a small body in a SRP dominated environment, we find that SRP acting alone will cause an initially circular orbit to precess around the pole of the primary's heliocentric orbital plane. When the gravitational and non-gravitational perturbations act in concert, the resulting equilibrium planes turn out to be qualitatively different, in some cases, from those obtained without considering the radiation pressure. The warping of the surfaces swept out by the modified equilibria as the semi-major axis varies depends critically on the cross-sectional area of the body exposed. These results, together with an adiabatic invariance argument on Poynting–Robertson drag, provide a natural qualitative explanation for the initial albedo dichotomy of Saturn's moon, Iapetus.

The classical Laplace plane as a stable disposal orbit for geostationary satellites

The classical Laplace plane is a frozen orbit, or equilibrium solution for the averaged dynamics arising from Earth oblateness and lunisolar gravitational perturbations. The pole of the orbital plane of uncontrolled GEO satellites regress around the pole of the Laplace plane at nearly constant inclination and rate. In accordance with Friesen et al. (1993), we show how this stable plane can be used as a robust long-term disposal orbit. The current graveyard regions for end-of-life retirement of GEO payloads, which is several hundred kilometers above GEO depending on the spacecraft characteristics, cannot contain the newly discovered high area-to-mass ratio debris population. Such objects are highly susceptible to the effects of solar radiation pressure exhibiting dramatic variations in eccentricity and inclination over short periods of time. The Laplace plane graveyard, on the contrary, would trap this debris and would not allow these objects to rain down through GEO. Since placing a satellite in this inclined orbit can be expensive, we discuss some alternative disposal schemes that have acceptable cost-to-benefit ratios.

DYNAMICAL INSTABILITIES IN HIGH-OBLIQUITY SYSTEMS

High-inclination circumplanetary orbits that are gravitationally perturbed by the central star can undergo Kozai oscillations---large-amplitude, coupled variations in the orbital eccentricity and inclination. We first study how this effect is modified by incorporating perturbations from the planetary oblateness. Tremaine et al. (2009) found that, for planets with obliquities > 68.875 degrees, orbits in the equilibrium local Laplace plane are unstable to eccentricity perturbations over a finite radial range, and execute large-amplitude chaotic oscillations in eccentricity and inclination. In the hope of making that treatment more easily understandable, we analyze the problem using orbital elements, confirming this threshold obliquity. Furthermore, we find that orbits inclined to the Laplace plane will be unstable over a broader radial range, and that such orbits can go unstable for obliquities less than 68.875 degrees. Finally, we analyze the added effects of radiation pressure, which are important for dust grains and provide a natural mechanism for particle semimajor axes to sweep via Poynting-Robertson drag through any unstable range. We find that generally the effect persists; however, the unstable radial range is shifted and small retrograde particles can avoid the instability altogether. We argue that this is occurs because radiation pressure modifies the equilibrium Laplace plane.

INFLUENCE OF THE SPOT-SIZE AND CROSS-SECTION ON THE OUTPUT FIELDS AND POWER DENSITY ALONG THE STRAIGHT HOLLOW WAVEGUIDE

This paper presents a rigorous approach for the propagation of electromagnetic (EM) flelds along a straight hollow waveguide with a circular cross section. The objectives are to present the technique to calculate the dielectric proflles and their transverse derivatives in the inhomogeneous cross section of the cylindrical hollow waveguide, and to understand the in∞uence of the spot-size and cross section on the output flelds and output power density. The derivation is based on Maxwell's equations. The longitudinal components of the flelds are developed into the Fourier-Bessel series. The transverse components of the flelds are expressed as functions of the longitudinal components in the Laplace plane and are obtained by using the inverse Laplace transform by the residue method. The separation of variables is obtained by using the orthogonal-relations. These objectives contribute to the application of the model for the straight hollow waveguide.

Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

We study the dynamics of the 3D three-body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole Hamiltonian expansion (third-order theory in the ratio of the semimajor axes), as third-order terms are needed to consider the secular variations of the outer perturber and potential secular resonances between the arguments of the pericentre and/or longitudes of the node of both bodies. Short-period averaging and node reduction (by adoption of the Laplace plane reference frame) reduce the problem to two degrees of freedom. The 4D dynamics is analysed through representative planes which identify the main equilibria of the problem. As in the circular problem (i.e. perturber on a circular orbit), the ‘Kozai-bifurcated’ equilibria play a major role in the dynamics of an inner body on a quasi-circular orbit: its eccentricity variations are very limited for mutual inclination between the orbital planes smaller than ∼40°, while they become large and chaotic for higher mutual inclination. Particular attention is also given to a region around 35° of mutual inclination, detected numerically by Funk et al. and consisting of long-time stable and particularly low-eccentricity orbits of the small body. Using a 12th-order Hamiltonian expansion in eccentricities and inclinations, in particular its action-angle formulation obtained by Lie transforms from Libert & Henrard, we show that this region presents an equality of two fundamental frequencies and can be regarded as a secular resonance. Our results also apply to binary star systems where a planet is revolving around one of the two stars.

Öpik-type collision probability for high-inclination orbits

The classical Öpik theory provides an estimate of the collision probability between two bodies on bound, heliocentric or planetocentric orbits under restrictive assumptions of: (i) constant eccentricity and inclination, and (ii) uniform circulation of the longitude of node and argument of pericenter. These assumptions are violated whenever either of the orbits has a large inclination with respect to the local Laplace plane or large eccentricity, and their motion is perturbed by an exterior (tidal) gravitational field of a planet or the Sun. In this situation, known as the Lidov–Kozai regime, the eccentricity and inclination values exhibit large and correlated oscillations. At the same time, the longitude of node and the argument of pericenter may have strongly nonlinear time evolution, with the latter being even bound to a small interval of values. Here we develop a new Öpik-type collision probability theory which is valid even for highly inclined and/or eccentric orbits of the projectile. We assume that the orbit of the target is circular and in the local Laplace plane. Such a generalized setting is necessary, as an example, to correctly estimate the terrestrial impact fluxes of sporadic micrometeoroids on high-inclination orbits (notably those from the toroidal source and the associated helion and anti-helion arcs).

Analysis of the applicability of the integral equation method in the theory of transient electroanalytical experiments for homogeneous reaction–diffusion systems: The case of planar electrodes

The method of integral equations (IEs) is a classical approach to formulating theoretical models of transient experiments in electroanalytical chemistry. In spite of many examples of such integral equations, the literature does not seem to contain any general discussion of the applicability of the method to homogeneous reaction–diffusion partial differential equations (PDEs). Such an analysis has been performed in this work, under assumptions of planar electrode, one-dimensional semi-infinite spatial domain, and all diffusion coefficients either distinct or identical. It is shown that in the distinct diffusion coefficient case the eigenvalues of the matrix of the system of ordinary differential equations (obtained from the Laplace transformation of the PDEs) can be separated in the Laplace plane, which facilitates the derivation of the IEs. However, the IEs are found to be easily obtainable only in the case of sufficiently simple networks of homogeneous reactions, for which the matrices of rate constants are appropriately sparse. Further work is needed to develop mathematical and numerical techniques enabling the application of the IE method to complex reaction networks.

FLEXIBLE HOLLOW WAVEGUIDE WITH TWO BENDINGS FOR SMALL VALUES OF STEP ANGLES, AND APPLICATIONS

This paper presents an improved approach for the propagation of electromagnetic (EM) flelds in the case of the ∞exible hollow waveguide that consists of two bendings in the same direction. In this case, the objective is to develop a mode model for infrared (IR) wave propagation along the ∞exible hollow waveguide, in order to provide a numerical tool for the calculation of the output flelds, output power density and output power transmission. The main steps of the method for the two bendings will introduced in the derivation, in detail, for small values of step angles. The derivation for the flrst section and the second section of the waveguide with the two bendings is based on Maxwell's equations. The separation of variables is obtained by using the orthogonal-relations. The longitudinal components of the flelds are developed into the Fourier-Bessel series. The transverse components of the flelds are expressed as functions of the longitudinal components in the Laplace plane and are obtained by using the inverse Laplace transform by the residue method. This model can be a useful tool in all the cases of the hollow toroidal waveguides, e.g., in medical and industrial regimes.

ERRATUM: “DETERMINING TITAN'S SPIN STATE FROM CASSINI RADAR IMAGES” (2008, AJ, 135, 1669)

We previously reported an initial determination of Titan's rotational state from fits to overlapping radar images. We have since discovered a coding error in software used to make these fits, which led to systematic offsets of 1-2 km in recovered positions. While our principal results remain qualitatively unchanged, with this error corrected, the pole movement we previously reported (our weakest result) is now counterindicated. Our revised best fit is essentially the same as the no pole result discussed at the top of the second column on page 1675. The determined pole location did not change significantly after the bug fix and thus we still conclude that the spin axis is not in the plane formed by Titan's orbit normal and the normal to the Laplace plane. Due to the correlations between pole wobble and spin rate (see Figure 3 on page 1672), the new best fit has a spin rate that differs from the synchronous value by an amount that is three times smaller than the value reported in the paper. The pole location changed by less than 0.01 deg (~500 m on the surface) and rate of increase in spin decreased by a factor of 2 from the previous fit. The new best-fit parameter values with 1σ error bars are: pole right ascension 39.4934 ± 0.0249 deg, pole declination 83.4368 ± 0.0024 deg, spin rate 22.57731 ± 0.00011 deg/day (0.00033 deg/day greater than synchronous spin rate), derivative of pole right ascension –6.52 ± 4.20 deg/century, derivative of pole declination –0.2212 ± 0.3567 deg/century, and derivative of spin rate 0.0247 ± 0.0050 deg/day/century. The corrected version of Table 3 (below) shows the residual systematic and random error of the several candidate models discussed in the paper. Fixing the bug reduced the residual systematic error of all the fitted models. The four models in which spin rate is allowed to vary from synchronous either due to a change in spin rate (Column 5, numbered from the left) or a change in its time derivative (Column 6) or both (Columns 7 and 8) have lower residual systematic errors and thus better represent the data than do the purely synchronous fit (Column 3). For this reason, an asynchronous spin rate is still supported by the data, although efforts (e.g., Mitchell 2009) to quantitatively interpret the asynchroneity should take our revised determination into account. On the other hand, as depicted by Columns 7 and 8, allowing the pole movement terms to vary from the predicted (IAU Titan) values results in no significant improvement in the fit, thus large short-term pole movement is not supported by the data. In fact, the best-fit values and error bars for the pole movement are consistent with the long-term pole trends that were predicted prior to the Cassini mission.

On the singularities of fractional differential systems, using a mathematical limiting process based on physical grounds

Fractional systems are associated with irrational transfer functions for which nonunique analytic continuations are available (from some right-half Laplace plane to a maximal domain). They involve continuous sets of singularities, namely cuts, which link fixed branching points with an arbitrary path. In this paper, an academic example of the 1D heat equation and a realistic model of an acoustic pipe on bounded domains are considered. Both involve a transfer function with a unique analytic continuation and singularities of pole type. The set of singularities degenerates into uniquely defined cuts when the length of the physical domain becomes infinite. From a mathematical point of view, both the convergence in Hardy spaces of some right-half complex plane and the pointwise convergence are studied and proved.

Determination of an instantaneous Laplace plane for Mercury’s rotation

Mercury is the target of two space missions: MESSENGER, which carried out its first and second flybys of Mercury on January 14, 2008 and October 6, 2008, and the ESA/JAXA space mission BepiColombo, scheduled to arrive at Mercury in 2020. The preparation of these missions requires a good knowledge of the rotation of Mercury. This paper presents studies performed by the MORE/ROMEO Team of BepiColombo mission. In particular, we show here an analytical study of the rotation of Mercury, and the determination of a suitable Laplace plane that can be considered as a reference inertial plane for the duration of the mission. Our main results are a prediction of the short-period perturbations that should be detected in the longitudinal rotational motion of Mercury, and an easy method to define an instantaneous Laplace plane taking into account the very small precession of Mercury’s orbit. This plane is afterward chosen as constant reference plane. An estimation of the (small) error induced by this choice is also computed.

Kozai resonance in extrasolar systems

Aims. We study the possibility that extrasolar two-planet systems, similar to the ones that are observed, can be in a stable Kozai-resonant state, assuming a mutual inclination of the orbital planes of order .Methods. Five known multi-planet systems that are not in mean motion resonance were selected, according to defined criteria, as “possible prototypes” (υ Andromedae, HD 12661, HD 169830, HD 74156, HD 155358). We performed a parametric study, integrating several sets of orbits of the two planets, obtained by varying the (unknown) inclination of their orbital planes and their nodal longitudes, thus changing the values of their masses and mutual inclination. We also take into account the reported observational errors on the orbital elements. These numerical results are characterized using analytical secular theory and frequency analysis. Surface of section techniques are also used to distinguish between stable and chaotic motions. Results. Frequency analysis offers a reliable way of identifying the Kozai resonance in a general reference frame, where the argument of the pericenter of the inner planet does not necessarily librate around ± as in the frame of the Laplace plane, through the non-coupling of the eccentricities of the two planets. We find that four of the five selected systems (υ Andromedae, HD 12661, HD 169830 and HD 74156) could in principle be in Kozai resonance, as their eccentricities and apsidal orientations are such that the system enters in the stability region of the Kozai resonance in 20-70% of the cases, provided that their mutual inclination is at least . Thus, a large fraction of the observed multi-planet systems has observed orbital characteristics that are consistent with stable, Kozai-type, motion in 3D. Unstable sets of orbits are also found, due to the chaos that develops around the stability islands of the Kozai resonance. A variety of physical mechanisms that could generate the necessary large mutual inclinations are discussed, including (a) planet formation; (b) type II migration and resonant interactions during the gas-dominated phase; (c) planetesimal-driven migration and resonance crossing during the gas-free era; (d) multi-planet scattering, caused by the presence of an additional planet.

Secular frequencies of 3-D exoplanetary systems

Using a 12th order expansion of the perturbative potential in powers of the eccentricities and the inclinations, we study the secular effects of two non-coplanar planets which are not in mean–motion resonance. By means of Lie transformations (which introduce an action–angle formulation of the Hamiltonian), we find the four fundamental frequencies of the 3-D secular three-body problem and compute the long-term time evolutions of the Keplerian elements. To find the relations between these elements, the main combinations of the fundamental frequencies common to these evolutions are identified by frequency analysis. This study is performed for two different reference frames: a general one and the Laplace plane. We underline the known limitations of the linear Laplace–Lagrange theory and point out the great sensitivity of the 3-D secular three-body problem to its initial values. This analytical approach is applied to the exoplanetary system \({\upsilon}\) Andromedae in order to search whether the eccentricities evolutions and the apsidal configuration (libration of \({\Delta \varpi}\)) observed in the coplanar case are maintained for increasing initial values of the mutual inclination of the two orbital planes.

Multimodal analysis of force spectroscopy based on a transfer function study ofmicro-cantilevers

Quantitative force spectroscopy experiments require a comprehensive knowledge of thefrequency response characteristics of the micro-cantilever. To establish a generic theoreticalmodel an analytical mathematical description of the cantilever dynamics in forcespectroscopy was developed using a transfer function approach. This study allows forquantitative evaluation of force spectroscopy experiments and analysis of stability andcontrollability of an atomic force microscope (AFM). The model accounts for the dynamiccharacteristics of the extended cantilever beam and for elastic sample properties. Thesystem dynamics were investigated using an exact system-theoretic approach. The stepand frequency responses are given for force spectroscopy experiments in differentexperimental configurations. The transfer function approach used in this studyallows us to investigate very significant dynamic aspects that simple first modeapproximations cannot capture. Only extended beam models account for both poles andzeros of the transfer function and can thus reproduce important features that arerelated to the zero dynamics. These features include pole-zero cancellations ornon-minimum phase response. The possibility of non-minimum phase response in AFM ishighly important for the design of inverse filters. The presence of zeros in theright half of the Laplace plane immediately implies that the inverse system isunstable.

Subsurface Sensing Under Sensor Positional Uncertainty

We consider the problem of classifying buried objects using electromagnetic induction data collected in a setting where there are errors in sensor positioning. Using a series of decay constants (or equivalently, Laplace plane poles) as features for classification, our algorithm seeks to estimate these poles and, subsequently, to determine the type of object in the sensor field of view. In many practical scenarios, a set of data is often accompanied by domain knowledge that the location of the transmitters and/or receivers is only known to within some degree of accuracy (e.g., 10 cm in the along-track direction and 5-cm cross-track). Here, we develop an approach to the extraction of information from such data sets in which the quantitative positional bound information is used in the context of a min-max optimization strategy. Specifically, we look for the parameters of interest that minimize the maximum data residual, where the maximum error is computed over ellipsoids or polyhedra of possible sensor locations defined by the bound information. Our formulation admits data collection with independent or dependent positional uncertainty values at successive nominal collection locations. Our algorithms for solving this optimization problem are validated using simulated and measured data

The application of flow dispersion models to the FM01-LC laboratory filter-press reactor

The flow distribution in the rectangular channel of a laboratory filter-press electrochemical reactor was evaluated using three flow models namely: (a) axial dispersion, (b) sum of two phases and (c) fast and stagnant zones. In the case of the axial-dispersion model, several methods have been used to calculate the Peclet number; the moment method, the non-linear least-squares and the Laplace transform technique. Several boundary conditions, involving different physical and experimental assumptions of the flow were used to solve the partial differential equation that describes the flow behaviour. A total of nine expressions to examine flow dispersion has been used. The comparison of experimental and predicted response signals was made by evaluating the root mean squared error. A data fit in real time has been found to be a better choice as solutions based on the evaluation of moments are prone to error due the overweight of the signal at long times. Data fitting in the Laplace plane is very accurate but it does not guarantee a good fit in real time. Models based on the sum of a fast and a slow or stagnant phase resulted in solutions having very low values of the extension of the slow and stagnant phases, the assumption of a single phase with some degree of dispersion was considered more appropriate.