Literal Solution Research Articles

Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.

The Yarkovsky Seasonal Effect on Asteroidal Fragments: A Nonlinearized Theory for Spherical Bodies

The long-term orbital evolution of meteoroids and small asteroids in the size range up to several kilo- meters is aUected by the ii seasonal ˇˇ Yarkovsky force, caused by radiation pressure recoil on spinning bodies heated by solar radiation to diUerent temperatures at diUerent latitudes on their surfaces. This eUect leads to a draglike, secular semimajor-axis decay, which may inject the bodies into chaotic zones associated with mean motion and secular resonances and eventually deliver them to near-Earth space. To model the Yarkovsky force, two kinds of simplifying assumptions have been frequently made: (1) a linearization in the ratio between temperature variation and average temperature and in the orbital eccentricity and (2) a plane-parallel geometry, that is, body sizes larger than the penetration depth of the seasonal thermal wave (typically, several meters). In a previous paper, we developed a new nonlinearized theory, and here we also remove the assumption of plane-parallel geometry and extend our theory to the more general spherical case, valid for all body sizes, by means of a new numerical approach. We also revise the linear theory, obtaining a fully analytical literal solution, which is well suited to develop ana- lytical and semianalytical secular perturbation theories and allows us to perform a detailed comparison with the results of the more accurate numerical model. We —nd that the accuracy of the linear theory is relatively good (20% or better) for near-circular orbits. Although the temperature variations grow with the orbital eccentricity, we show that the linear theory can still predict the averaged drift rates of the mean orbital elements up to eccentricities of 0.4¨0.5.

Literal solutions for time-domain crosstalk on lossless transmission lines

The transmission line equations for a three-conductor lossless line in a homogeneous medium are solved in literal (symbolic) form in the time domain. The resulting formulas for the crosstalk voltages are exact and are given in terms of the line parameters and the termination impedances. These formulas demonstrate the effect of the various line parameters on the resulting crosstalk and show how to adjust the line parameters to achieve a desired time-domain crosstalk. In addition, other solution methods are discussed and used to verify the literal solution. >

Analytical theory of motion of phoebe, the ninth satellite of saturn

The literal solution of the restricted three body problem obtained by the authors up to the eleventh order with respect to the minor parameter is applied to the investigation of the motion of Phoebe, the ninth satellite of Saturn. As distinct from the existing analytical theories of the motion of the satellite, in the present paper the planetary perturbations are taken into account. A comparison with the modern numerical theory of the motion of Phoebe has shown that the new analytical theory of the satellite motion represents observations with the same degree of accuracy.

Theorie litterale du neuvieme satellite de Saturne, Phoebe

We study a theory for the ninth satellite of Saturn, Phoebe, based on the literal solution we have obtained in the main problem of the lunar theory.

Literal solution for Hill's lunar problem

A computer program for the manipulation of power series in several variables is used to find the first order solution to Hill's lunar problem. The ratiom of the mean motion of the Sun to that of the Moon is kept as a formal parameter. The series inm are known to converge very poorly. It is shown how efficient algorithms among them the Lie transformation allow us to compute the series inm as far as they are needed. When the series are evaluated at Brown's numerical value form the results achieve or exceed his accuracy.

On some approximations in Brown's lunar theory

A re-evaluation of Brown's Lunar Theory has long been awaited. While working on this problem a number of questions about the adequacy of Brown's approach have arisen, and some of these questions including that of the need for a literal solution of the main problem are discussed in this paper.

ANALYSIS OF A GENERAL POPULATION GENETIC MODEL OF MEIOTIC DRIVE.

The explicit algebraic analysis of any population model which takes non-Mendelian segregation into account is generally a cumbersome affair because the frequency of the gene which causes the abnormal segregation is not the same in male and female gametes. Models of this kind generate a messy set of simultaneous nonlinear equations from which the equilibria of gene frequency may be extracted. In specific applications it is often sufficient to solve the equations numerically, behaving as if they had no literal solution. This procedure is inelegant, and it may lead to some practical problems. If an equilibrium cannot be found for a given set of parameters, for example, then one has to try to decide if there is in fact no equilibrium for the parameters or if the iterative procedure happens to diverge for that particular set. When the analytic roots are available this problem never arises. The purpose of this article is to present the detailed solution of a model of meiotic drive which Lewontin (1968) has suggested would be helpful in understanding the evolutionary dynamics of the t-alleles in the house mouse. Because mice tend to breed in small endogamous family units, however, the deterministic model is only a very rough approximation to what would be expected to occur in nature, and to gain better insight one is forced to undertake Monte Carlo simulation of small populations (Lewontin and Dunn, 1960; Lewontin, 1968). Another striking example of non-Mendelian segregation occurs in the segregation distorter system in Drosophila melanogaster. Since local Drosophila populations are at least one and probably two orders of magnitude larger than local mouse populations (Wright et al., 1942), stochastic fluctuations in gene frequency are of minor importance as compared with the effects of selection and meiotic drive. Therefore, the deterministic model is probably a reasonable approximation to Drosophila populations, as well as to other large populations in which non-Mendelian segregation of an autosomal gene occurs (such as wheat, Loegering and Sears, 1967). The model to be analyzed is actually more general than the one proposed by Lewontin (1968) as this one allows for arbitrary segregation ratios in the two sexes. One disappointment is that the equilibria turn out to be quite complicated. On the other hand, the analysis does lead to a remarkably simple criterion for predicting when nontrivial gene frequency equilibria will occur and for determining the stability or instability of these equilibria.

The literal solution of the main problem of the lunar theory

The literal solution of the main problem of the lunar theory