Some problems in Project "Fobos-Ground"

R.Z.Akhmetshin, G.B.Efimov

Keldysh Institute of Applied Mathematics, RAS

Miusskaya pl., 4, Moscow, 125047, Russia

 

For many years the calculations of optimal trajectories of spacecraft (SC) with low thrust (LT) towards small bodies of Solar system are carried out in Keldysh Institute of Applied Mathematics. The scientific importance of small celestial bodies' investigation, especially of Main Belt asteroids, Jupiter group comets and satellites of big planets, is determined by fundamental scientific problem - the origin and evolution of Solar system. The necessity of elaborate study of mineralogical and isotopic composition of small bodies raises the problems of landing on them and consequent deliver of ground sample to the Earth. Both Russia-German Project "Fortuna" and Russian "Fobos-Ground" aim to deliver the "relic" matter from small bodies with the use of low thrust engines. The latter one is included into Russian Federal Space Program. It is supposed to reach Fobos, or more exactly - the gravisphere of Mars, with the use of spacecraft with electro-rocket engines and solar arrays.

љFor solution of a number of questions at the first stage of investigation of optimal trajectories to Fobos the model of "ideally controlled" LT, without constraints on the value and direction of jet thrust and on the value of jet velocity, was used. The model with constant jet velocity V_J = const is more realistic from the technical point of view but it is more difficult for trajectory optimization and for analysis of the problem on the whole in comparison with ideal one. As in many cases both models of LT gives rather similar results it's possible to use the ideal model as more simple and easy for calculations.

For the ideal model of LT with constant power the optimization problem can be divided into two separate problems: that is the optimization of the trajectory can be reduced to minimization of so called "energy consumption performance index" J, and it can be done independently of the choice of spacecraft and propulsion system parameters (i.e. optimum distribution of the system masses). Since the value of J being defined then the final mass of spacecraft and propellant mass can be easily obtained for different parameters of SC and propulsion system by formulae. In the case of solar arrays with power dependant upon the distance from the Sun a modified index J must be used.

The flight duration from Earth to Mars may vary in the range from about one year to a little more than two years. So one have to calculate and analyze a set of optimal trajectories for these range of flight duration and for various values of other substantial parameters of the problem, such as parameters of SC, LT and HT engines and so on. The model with V_J = const makes necessary to consider different values of V_J and of hyperbolic geocentric excess velocity V_¥, lengths and quantity of pauses in the work of engines, dates of departure and arrival. The full optimization of all these parameters makes difficult the conceptualization of numerous calculations and analysis of concrete factor's influence. The use of ideal LT model in this case may give considerable advantages.

For the start date t_S towards Mars of launcher "Souz" and SC with low thrust in 2005 year two pictures of isolines for the final mass of SC near Mars, as functional н_F(t_S,ф),љ were built for the range ф = 250-850 days of flight duration with a fixed value V_¥ = 1 km/s and with optimal value V_¥. They gave an expanded overview of the problem in whole; particularly they helped to understand why is it rather difficult to get an additional gain in the final mass with the only increase of flight time.

It's also noted that in the case of low thrust model with V_J = const the picture of isolines changes slightly and difference in the final SC mass for two models is small.