Long-term variations of the Earth's orbital elements

Critical analysis of theories of the long-term variations of the ecliptical elements of the Earth leads to the following conclusions, regarding the influence of different terms on the accuracy of the expansions used:

1. further improvement in planetary masses will not have significant influence:
2. for the (e, π) system, terms depending upon the second order as to the disturbing masses are more important than ones coming from the third degree with respect to the planetary eccentricities and inclinations;
3. for the (i, Ω) system, the latter terms have highly significant influence, whereas additional terms in masses are negligible. The same conclusion can be drawn for (ε,Ψ _g ). Using these results, a new solution for the long-term variations of the Earth's orbital elements is obtained. The results fore, π,i, Ω include terms depending upon the second power as to the disturbing masses and to the third degree with respect to the planetarye's andi's. For the obliquity ε and the annual general precession in longitudeΨ _g , a Laplace series is proposed where amplitudes, mean rates and phases are computed from those of the (i, Ω) system.

     

Periodic solutions in the commensurable three-body problem

Various families of periodic solutions are shown to exist in the three body problem, in which two of the bodies are close to a commensurability in mean motions about the third body, the primary, which is considerably more massive than the other two. The cases considered are

1. The non-planar circular restricted problem (in which one of the secondary bodies has zero mass, and the other moves in a fixed circular orbit about the primary).
2. The planar non-restricted problem (in which the three bodies move in a plane, and both secondaries have finite mass).
3. The planar elliptical restricted problem (in which the three bodies move in a plane, one of the secondary bodies has zero mass, and the other moves in a fixed elliptical orbit about the primary).

The method used is to eliminate all short period terms from the Hamiltonian of the motion by means of a von Zeipel transformation, leaving only the long period terms which are due to the commensurability. Hence only the long period part of the motion is considered, and the variables used differ from the variables describing the full motion by a series of short-period trigonometric terms of the order of the ratio of the mass of the secondaries to that of the primary body. It is shown that solutions of the long-period problem in which the variables remain constant are equivalent to solutions in the full motion in which the bodies periodically return to the same configuration, and these are the types of periodic solution that are shown to exist. The form of the disturbing function, and hence of the equations of motion, is found up to the fourth powers of the eccentricities and inclination by considering the d'Alembert property. The coefficients of the terms appearing in this expansion are functions of the semi-major axes of the orbits of the secondary bodies. Expressions for these coefficients are not worked out as they are not required. Lete, n, m be the orbital eccentricity, mean motion and mass of one of the secondary bodies, and lete′, n′, m′ be the corresponding quantities for the other. (The mass of the primary is taken as unity). In cases (a) and (c) we will havem=0. In case (a)e′ will be zero, and in case (c) it will be a constant. Leti be the mutual inclination of the orbits of the secondary bodies. Suppose the commensurability is of the form(p+q) n =pn′, wherep andq are relatively prime integers, and put γ=(p+q) n/n′?p. The families of periodic solutions shown to exist are as follows. For q=1 No periodic solutions are found withi≠0 in case (a), and none withe′≠0, in case (c). In case (b) periodic solutions are found in whiche=0 (m′/γ),e′=0 (m/γ) for values of γ away from the exact commensurability. As γ approaches zero thene ande′ become 0 (1). For q≠1 Case (a). Families of periodic solutions bifurcating from the family withe=0, i=0 are shown to exist. Families in whichi=0 ande becomes non-zero exist for all values ofq. Families in whiche=0 andi becomes non-zero exist for even values ofq. Families in whiche andi become non-zero simultaneously exist for odd values ofq. Case (b). No families are found other than those withe=e′=0. Case (c). Families are found bifurcating from the familye=e′=0 in whiche ande′ become non-zero simultaneously. For all these solutions existence is only demonstrated close to the point of bifurcation, where all the variables are small, as the method uses series expansions ine, e′ andi. From the form of the solutions it is clear that the non-zero variables will become large for values of γ away from the bifurcation point.     

The influence of the great inequality on the secular disturbing function of the planetary system

This paper derives the contributionF ₂ ^* by the great inequality to the secular disturbing function of the principal planets. Andoyer's expansion of the planetary disturbing function and von Zeipel's method of eliminating the periodic terms is employed; thereby, the corrected secular disturbing function for the planetary system is derived. An earlier solution suggested by Hill is based on Leverrier's equations for the variation of elements of Jupiter and Saturn and on the semi-empirical adjustment of the coefficients in the secular disturbing function. Nowadays there are several modern methods of eliminating periodic terms from the Hamiltonian and deriving a purely secular disturbing function. Von Zeipel's method is especially suitable. The conclusion is drawn that the canonicity of the equations for the secular variation of the heliocentric elements can be preserved if there be retained, in the secular disturbing function, terms only of the second and fourth order relative to the eccentricity and inclinations.The Krylov-Bogolubov method is suggested for eliminating periodic terms, if it is desired to include the secular perturbations of the fifth and higher order in the heliocentric elements. The additional part of the secular disturbing functionF ₂ ^* derived in this paper can be included in existing theories of the secular effects of principal planets. A better approach would be to preserve the homogeneity of the theory and rederive all the secular perturbations of principal planets using Andoyer's symbolism, including the part produced by the great inequality.     

Asteroids with large secular orbital variations

As the classical linear theory of secular perturbations for asteroids is known not to be adequate for computing the perturbations of asteroids with high eccentricities and/or inclinations, a seminumerical method to calculate the secular perturbations by including higher-degree terms in the disturbing function has been developed. It is here applied to asteroids with small values of $\text{(1 ? e}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{cos}\text{i}$, since the secular variations as well as their deviations from the results derived by the classical linear theory are generally large for such asteroids. It is found that the arguments of perihelion for five of the numbered asteroids are librating around 90 or 270°. For asteroids with $\text{(1 ? e}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>cos</mtext><mtext>\; i</mtext>}}$ less than 0.85 the results of the secular variations are tabulated. Also the stability of such orbits is discussed by comparing the orbital properties of short-periodic comets with them. Generally speaking, orbits of the asteroids are more stable than those of the short-periodic comets, and asteroids with librating arguments of perihelion are more stable than those with circular coplanar orbits although their orbital elements are changed more by secular perturbations.     

Invariant rotational curves in Sitnikov's Problem

The Sitnikov's Problem is a Restricted Three-Body Problem of Celestial Mechanics depending on a parameter, the eccentricity,e. The Hamiltonian,H(z, v, t, e), does not depend ont ife=0 and we have an integrable system; ife is small the KAM Theory proves the existence of invariant rotational curves, IRC. For larger eccentricities, we show that there exist two complementary sequences of intervals of values ofe that accumulate to the maximum admissible value of the eccentricity, 1, and such that, for one of the sequences IRC around a fixed point persist. Moreover, they shrink to the planez=0 ase tends to 1.     

Evolutionary considerations involving the internal density concentration parameter of binary stars

The evolutionary changes that occur in the internal density concentration parameterk ₂ (called the apsidal constant for brevity) for a star of given mass and initial composition are examined in detail. The purpose is to ascertain whether or not such an approach leads to a reduction in the differences now noted between the theoretically derived values ofk ₂ and the observed values derived from the secular advance of the periastron in close eclipsing binary systems.A series of stellar models of mass 2.0, 5.0, 10.0 and 20.0M were employed, with an initial compositional mixture ofX=0.739,Y=0.24 andZ=0.021. These models cover an evolutionary range from a point in time where the star has just completed the Hayashi phase of its pre-Main Sequence contraction through its entire Main Sequence phase to the point where hydrogen depletion in the core is complete.For each model, a value ofk ₂ is determined by numerically integrating Radau's equation and using the values of , the ratio of the star's density at pointa to its mean density, as taken from the models. The result is the time history ofk ₂ for each stellar mass over the evolutionary range of interest. The results are then summarized in the (logk ₂, logT _e) plane which, for the first time, quantitatively indicates the variation ink ₂ as a function of the evolutionary state of the star.A comparison between these theoretically derived values ofk ₂ and a selected set of observationally determined ones immediately indicates that the secular variation ink ₂ plays an extremely important part in any comparison between theory and observation. For most of the cases studied, the difference between the theoretically and observationally determined values ofk ₂ can be reconciled in terms of the evolutionary history of the binary system.While tentatively providing a satisfactory explanation for the previously noted differences in the determination ofk ₂, there now exists the problem of accurately pinpointing the evolutionary state of the observed binary system.     

A model of impulsive loop flares revisited

A critical examination of the components of the recent impulsive loop flare model of Takakura is made. It is found that his analysis of the stability of the electron distribution resulting from anomalous heat conduction is in error and electron plasma waves would not be excited. Rather, in the regions where the electron/proton temperature ratioT _e/T _i 10, electrostatic ion-cyclotron waves would be excited and in the regions whereT _e 10, ion-acoustic waves would be excited. Ratios ofT _e/T _i 10 occur only in the late time development behind the conduction fronts. Since the anomalous resistivity due to electrostatic ion-cyclotron waves is fortuitously about 70% of the one used by Takakura, the general development will follow closely the one calculated by him. Because the anomalous resistivity due to ion-acoustic waves is about 95 times the one used by Takakura, the development in the parts of the loop whereT _e/T _i 10 for late times would be altered considerably.Also Guest Worker at NOAA Space Environment Laboratory, Boulder, Colorado, U.S.A.     

Expansion of the perturbing function of a satellite restricted four-body problem

A satellite four-body problem is the problem of motion of an artificial satellite of a planet in a region of the space where perturbations due to the gravitational field of the planet are of the same order as perturbations due to influences of two perturbing bodies. In this paper an expansion of the perturbing function into a Fourier series in terms of angular Keplerian elements ( _j , _j ,M _j :j=0,1,2) (designations are standard) is obtained taking into account a sharp commensurability of the typen/ ₀=(p+q)/p (n is the mean motion of the artificial satellite and ₀ is the angular velocity of rotation of the planet,p andq are integers).The coefficients of the Fourier series are the functions of the positional Keplerian elements (a _j ,e _j ,i _j ;j=0, 1, 2) (designations are standard) and, in particular, are series in terms ofe _j that, generally speaking, can be written out to an accuracy ofe _j ¹⁹ .The expansion obtained can be used for the construction of a semianalytical theory of motion of resonant satellites on the basis of conditionally periodic solutions of the restricted four-body problem.     

Evaluation of disturbances from solar radiation in orbital elements of geosynchronous satellites based on harmonics

Using specialized codes for the search of periodic and linear components we show that direct solar radiation leads to short-period variations of all the orbital elements of geosynchronous satellites. The variation period of the semimajor axis a, orbit inclination i and the longitude of the ascending node Ω is 1 day. Eccentricity e, the argument of perigee ω and the mean anomaly M vary with a period of 0.5 days. Direct solar radiation also leads to long-period variations in e, ω and M with a period of 1 year. The elements a, i and Ω undergo variations only in the amplitude of diurnal variations with a period of 1 or 0.5 years. Secular variability (linear components) are not detected. To obtain the initial value array of the orbital elements we used the Lagrange equations of perturbed motion in the form of a Gaussian with their subsequent integration via a special method of harmonics: the values of the derived orbital elements, obtained from the Lagrange equations, were presented through the periodic functions that are easy to integrate.     

Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

A recurrence relation for inclination functions

When the terms of the series expansion for the gravitational potential of the Earth are expressed in terms of the orbital elements of an arbitrary Earth satellite, the orbital inclination,i, appears in each, term as the argument of a function of inclination only. For the special case when the field is axi-symmetric, studied in an earlier paper, a recurrence relation was given for a normalized inclination function,A ^k l(i), with two parameters. The present paper gives a recurrence relation for a general normalized function,K ^k lm(i), with three parameters.     

The occurrence of high-order resonances and Kozai mechanism in the scattered disk

By means of numerical methods we explore the relevance of the high-order exterior mean motion resonances (MMR) with Neptune that a scattered disk object (SDO) can experience in its diffusion to the Oort cloud. Using a numerical method for estimate the strength of these resonances we show that high-eccentricity or high-inclination resonant orbits should have evident dynamical effects. We investigate the properties of the Kozai mechanism (KM) for non-resonant SDO's and the conditions that generate the KM inside a MMR associated with substantial changes in eccentricity and inclination. We found that the KM inside a MMR is typical for SDO's with Pluto-like or greater inclinations and is generated by the oscillation of ω inside the mixed (e,i) resonant terms of the disturbing function. A SDO diffusing to the Oort cloud should experience temporary captures in MMR, preferably of the type 1:N, and when evolving inside a MMR and experiencing the KM it can reach regions where the strength of the resonance drops and consequently there is a possibility of being decoupled from the resonance generating by this way a long-lived high-perihelion scattered disk object (HPSDO).     

The high-eccentricity libration theory revisted

Using the local asymmetric expansion of the disturbing function for the planar elliptic restricted three-body problem up to degree 1 ine ₁ we develop a small amplitude libration theory. We review the laws that characterize the mean-motion resonances of asteroids with Jupiter and we obtain other new laws. Special attention is paid to the second forced mode whose equations are reformulated and new consequences of this component are discussed. An analytical expression for the trajectories in the phase space is obtained. The predictions are compared with numerical experiments which confirm the results.     

On the temperature of the helium emission regions in the solar atmosphere

An analysis of the spectral distribution of intensity of the Hei recombination continuum is probably the only direct method for determination of the electron temperature of helium emission regions on the Sun. On the basis of data on the Hei Lyman continuum, obtained by Dupree and Reeves from OSO-4, the electron temperature of undisturbed helium regions is determined: T _e = = 12500 K. Such a low T _e value is a serious argument in favour of the predominant role of UV coronal radiation in the helium ionization on the Sun. Comparison of the Hei Lyman continuum data with results of observations of the 10830 line showed that the visible helium lines and Hei Lyman continuum are produced within the same regions of the undisturbed solar atmosphere at T _e = 12500 K.     

Periodic perturbations in close binary systems due to rotational distortion

This paper deals with the perturbations which rotation can produce to the orbital elements of a close binary system. The rectangular components,R, S andW of the disturbing accelerations due to rotation have been substituted to the Gauss form of Lagrange's planetary equations to yield the first order approximation. The results obtained are exact for any value of orbital eccentricity between the values 0<e<1 and for arbitrary inclinations of the rotational axes to the orbital plane.First and second order approximations are given for the special case when equators are coplanar to the orbit.     

Trigonometric series representations of the orbital inclination function

In this paper, new trigonometric series representations of the orbital inclination functionF _imp (i) in multiples of cosines or sines will be established for all possible values ofl, m, andp. For such representations, the literal analytical expressions and the recurrence formulae satisfied by their coefficients will be established. Moreover, an economic algorithm for the table formulation of these coefficients for the possible values ofl, m, andp is given. Finally, numerical examples of the representations forl=2(1)4;m=0(1)l;p=0(1)l are also included.     

Generation of a d.c. field by nonlinear electromagnetic waves in relativistic plasmas

A finite amplitude linearly polarized electromagnetic wave propagating in a relativistic plasma, is found to generate the longitudinal d.c. as well as the oscillating electric field at the second harmonic. In a plasma consisting of only electrons and positrons, these fields cannot be generated.The evolution of the electromagnetic waves is governed by the non-linear Schrödinger equation which shows that the electromagnetic solitons are always possible in ultra-relativistic plasmas (electron-ion or electron-positron) but in a plasma with relativistic electrons and nonrelativistic ions, these solitons exist only if 1(KT _e/m_eC²)<(2m _i/15m_e);m _e andm _i being the electron and ion mass andT _e the electron temperature. Both the d.c. electric field and the solitons provide a nonlinear mechanism for anomalous acceleration of the particles. This model has direct relevance to some plasma processes occurring in pulsars.     

On the Aerodynamic Drag Force Acting on Interplanetary Coronal Mass Ejections

It is well known that the interaction of an interplanetary coronal mass ejection (ICME) with the solar wind leads to an equalisation of the ICME and solar wind velocities at 1 AU. This can be understood in terms of an aerodynamic drag force per unit mass of the form F _D/M=−(ρ_e AC _D/M)(V _i−V _e)∣V _i−V _e∣, where A and M are the ICME cross-section and sum of the mass and virtual mass, V _i and V _e the speed of the ICME and solar wind, ρ_e the solar wind density, C _D a dimensionless drag coefficient, and the inverse deceleration length γ=ρ_e A/M. The optimal radial parameterisation of γ and C _D beyond approximately 15 solar radii is calculated. Magnetohydrodynamic simulations show that for dense ICMEs, C _D varies slowly between the Sun and 1 AU, and is of order unity. When the ICME and solar wind densities are similar, C _D is larger (between 3 and 10), but remains approximately constant with radial distance. For tenuous ICMEs, the ICME and solar wind velocities equalise rapidly due to the very effective drag force. For ICMEs denser that the ambient solar wind, both approaches show that γ is approximately independent of radius, while for tenuous ICMEs, γ falls off linearly with distance. When the ICME density is similar to or less than that in the solar wind, inclusion of virtual mass effects is essential.     

The equations of motion of an artificial satellite in nonsingular variables

The equations of motion of an artificial satellite are given in nonsingular variables. Any term in the geopotential is considered as well as luni-solar perturbations up to an arbitrary power ofr/r, r being the geocentric distance of the disturbing body. Resonances with tesseral harmonics and with the Moon or Sun are also considered. By neglecting the shadow effect, the disturbing function for solar radiation is also developed in nonsingular variables for the long periodic perturbations. Formulas are developed for implementation of the theory in actual computations.     

Periodic orbits around an oblate spheroid

The general properties of certain differential systems are used to prove the existence of periodic orbits for a particle around an oblate spheroid.In a fixed frame, there are periodic orbits only fori=0 andi near /2. Furthermore, the generating orbits are circles.In a rotating frame, there are three families of orbits: first a family of periodic orbits in the vicinity of the critical inclination; secondly a family of periodic orbits in the equatorial plane with 0<e<1; thirdly a family of periodic orbits for any value of the inclination ife=0.