Abstract:
The problem of constructing a system of four point masses, the totality of which is equal to a predetermined solid, is solved. A family of such systems has been built, depending on six parameters. The freedom to choose parameters makes it possible to set the task of finding the set of masses that best approximates the momentums of the distribution of masses of the third order of the body. The problem in this formulation is solved in relation to the nucleus of comet 67P/Churyumov–Gerasimenko. The criterion for the quality of the coincidence of the momentums of the third-order mass distribution is the standard deviation of the momentums of the point mass system from the corresponding momentums of the comet nucleus. A system of four material points is constructed, minimizing the value of the root-mean-square error. This value turned out to be less than the similar values obtained in previous studies. It is noteworthy that the masses of the found points turned out to be different from each other and all are located inside the core, so that the three smallest of them are located in a larger fraction of the core, and the fourth, which concentrates $\approx$ 28.5% of the total mass of the core, is located inside a smaller fraction. This mass distribution is in good agreement with the well-known estimate of 27% of the volume of a smaller fraction of the comet’s core from the total volume.
Key words:equimomental systems, dynamically equivalent systems, approximation of the gravitational field of a small celestial body, equilateral tetrahedron, higher-order inertia integrals, Euler–Poinsot tensor, comet 67P/Churyumov–Gerasimenko.
