The advantages and disadvantages of a solution show in the ratio between its cost and the quantity and value of the information provided as well as in the flexibility and adaptability of its use. Cost is calculated taking into account the time and amount of calculation required to obtain a given solution.
Generally speaking, obtaining analytical solutions is a very arduous task and it requires a great deal of calculation, although it can be carried out once and for all. As far as non-linear problems are concerned - i.e. perturbations - analytical solutions imply simplifications that slightly reduce the generality of the obtained solution and the exactness of the results. These - together with the above mentioned bulk of work - are two of the main defects of analytical solutions, which fail to be suitable in case high precision is required, especially on long integration periods. The analytical solution, even as a first approximation, enables one to directly read the characteristics of the motion and to draw general conclusions about the problem. Sometimes this may lead to a mistake because the approximations may make the solution incomplete thus hiding effects that could be significant to the dynamics of the system. Thus this kind of solution requires constant attention to the elements to be left out and to the appropriate troncations. The currently existing algorithms to obtain an analytical solution enable the achievement of ever-increasing precision through iterative procedures that gradually improve the solution. As a drawback, they involve an ever-increasing amount of algebra and a solution which becomes more and more complicated and difficult to handle.
A suitable analytical solution is a very powerful instrument also because of its quick application. These solutions are employed for remote sensing, control devices and as a starting point for further numerical developments; they can also substitute numerical propagators (Millani, 1995), thanks to their speed. In that case the analytical solution is considered to be semi-analytical because several parameters are obviously to be numerically defined. The quick application of analytical and semi-analytical solutions is due to their independence from the integration period. As a matter of fact, the work spent on obtaining an analytical solution and its use do not depend directly on the integration period, as opposed to numerical solutions. The solution at any time does not require a new integration: the state vector can be calculated for each instant without calculating the solution for intermediate instants, which is the case for a step-by-step numerical integration.
Numerical solutions - explicit or implicit - with step-by-step integration do not allow one to impose specific constraints to the solution one is looking for. Moving backwards and forwards in time, a specific solution is automatically selected and its characteristics can be analised afterwards. Firstly we proceeded to study the long period effects with Cook's linearised method. In order to eliminate the lack of generality of Cook's solution - due to linearisation - the same study has been carried out with a sheer numerical procedure.
This procedure has led to results which basically agree with those achieved by Cook's method and it has also revealed solutions and errors which couldn't be detected by the procedure of linearisation. This preliminary study of the long period effects was followed by a study of the short and medium period. To solve this problem, a method was sought for that was generally applicable and able to give solutions whose validity was not limited by a number of initial hypothesis. This method should have exploited part of the advantages of both analytical and numerical solutions. It should also have allowed the introduction of a-priori constraints to the search for solutions, so that the existence of practically or theoretically interesting families of solutions could be automatically verified.
The problem was thus re-formulated in a "weak" form through Hamilton's variational principle and a numerical procedure of calculation with Finite Time Elements was adopted. Here the problem has been dealt with in Cartesian co-ordinates to simplify the development of the potential function; however, we have shown the procedures to obtain the solution both in orbital elements and in Delunay's canonical elements.
The method does neither depend on the kind of co-ordinates nor on the type of potential of forces employed, it is a numerical method and therefore it enables one to achieve the required precision and accuracy without further effort.
To adopt the variational principle means to shift from a differential problem to a problem of constraint optimisation. This allows one to impose some constraints on the overall orbital behaviour (even if Cartesian co-ordinates are employed instead of Keplerian elements). These constraints can either be imposed on the value of the unknowns or they can be of the functional type; they do not have to correspond to physical constraints on the motion, they can however represent a characteristic of the family of solutions one wants to obtain. This method is implicit and conservative both if used to integrate step by step (it can be employed like an ordinary Runge-Kutta) and if the orbit is globally integrated with an assembling procedure which will be described afterwards.
Such a method does not require any sort of initial semplifying hypothesis and it provides both a numerical and, in some cases, an analytical solution as an explicit function of time (or of the equivalent parameter used for the integration). This method also provides a transitional matrix which will be useful afterwards to study the stability of the obtained solution and its dependence on the initial conditions. The FTE method is usually applied to solve several problems of mechanics, but literature gives us no evidence of theories testifying its use in the field of celestial dynamics. In many works the FTE method has been applied to solve problems of structural mechanics, multy-body mechanics and fluid dynamics. These works have been used as a starting point to study a way to adapt and set up a method to solve problems of celestial mechanics. To be more precise, we have adopted - as an example - the study of the perturbations induced by the gravitational field of the moon. The results obtained have a demonstrative validity and confirm the efficiency and flexibility of the FTE method as far as orbital mechanics is concerned.
In order to verify the exactness of the results achieved by the study of the long period effects, a numerical propagator - with Runge-Kutta of 5th order at a constant step - has been employed with the integration of Lagrange's planetary equations which had been averaged on a period and deprived of the medium period terms. The step for Runge-Kutta has been chosen after trying different solutions, going from one minute up to one day, and checking the error within a span of six weeks of integration. At a 2-hour step, which allows reasonable periods of integration, one commits an error below 0,1%. In order to verify the results achieved with the FTE method, a numerical propagator - with Runge-Kutta-Fehlberg of 4th\5th order at a variable step - has been employed, integrating the equation of motion in Cartesian co-ordinates both with Cowell's and with Enke's methods.
I should like to start by saying that my solution depends anyhow on the knowledge of the gravitational field, and it can be more or less reliable according to the validity of the data of the lunar field.
--- WinMMMR v1.00unr * Massimiliano Vasile - . - M.Vasile@agora.stm.it -.-phone: __39-2-4235999 -.-fax:__39-2-48951568