Modeling of gravitational fields of small bodies of the Solar System

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Importance of the topic. As of today, various methods are applied to study the gravitational fields of celestial bodies. One of the most widespread methods is the method of spherical harmonics, based on the fact that the gravity potential of a celestial body can be expanded in spherical functions if the assumption of the density homogeneity of a celestial body is true. The accuracy of the gravitational field model calculated by the method of spherical harmonics depends largely on the accuracy of the elevation model of the celestial body in question.

From the scientific investigations made by the Laboratory, it follows that the method of spherical harmonics can successfully be applied to the modeling of the gravitational field of small bodies of the Solar System, like Phobos.

However, spherical harmonics have a number of serious drawbacks, namely:

- Spherical harmonics are the global carrier functions ;

- The method of spherical harmonics yields the best result when there is an even distribution of control points;

- The method of spherical harmonics does not fit the building of local models of the gravitational field;

- The total of number of the terms in the gravity potential expansion into series of spherical harmonics may not agree with the accuracy needed of the potential assessment;

- A change of one of the control points leads as a result to the change of all coefficients of the gravitational field expansion into spherical harmonics series.

In order to overcome the above drawbacks inherent in spherical harmonic functions, we offer to use the multifractal approach to construction of multi-level (hierarchical) approximations of gravitational fields.

The said approach is based on the statement that geopotential fields are multifractal as they possess a hierarchically ordered self-similar structure that can be considered in the process of approximation. In the multifractal (zonal) presentation of the gravity potential of celestial bodies, it is resolved into low-frequency and high-frequency components with the use of spherical scaling and wavelet-functions acting as low-frequency and high-frequency filters. The multifractal approach offered to approximating the gravity potential with the application of spherical wavelets has a variety of advantages over the traditional approach using spherical harmonics. The main advantage of the multifractal approach is that it suits well the building of local models of the gravitational field and can be applied even when the network control points are distributed unevenly to the most degree. It is necessary to notice also that the approach offered can be used for assessment of the gravity potential in the vicinity of celestial bodies whose shape differs highly from the sphere or ellipsoid.

Research aims are to work out a theory and methods for modeling gravitational fields of small bodies of the Solar System, based on their zonal presentation by means of both multifractal and wavelet-methods as well as and the application of the methods developed to constructing multi-level models of the gravitational fields of Phobos and Deimos.

Research tasks:

1. To make a relative analysis of existing methods for modeling gravitational fields of small bodies of the Solar System.

2. To work out innovative methods for modeling gravitational fields of small bodies of the Solar System, based on their zonal representation by means of multifractal and wavelet-methods.

3. To work out a technology for a zonal modeling of gravitational fields of small bodies of the Solar System.

4. A construction of zonal multifractal models of the gravitational fields of Phobos and Deimos.

5. Research into the possibility of application of the developed models of the gravitational fields of the Martian moons to studying their inner structure and surface morphology as well as finding out their origin and evolution.

The theoretical and practical results obtained by the current moment:

1. Spherical harmonious models of both the surface and Newtonian potential of Phobos have been constructed.

2. The theoretical bases for modeling gravitational fields of small bodies of the Solar System, based on their zonal representation have been devised.

Further development in this area:

1. The research field

To work out the theory and methods for multifractal approximation of gravity fields of celestial bodies from remote sensing data.

2. The applied field

To work out working models of the gravitational fields of Phobos and Deimos.

3. The educational field

To develop training courses for students and specialists, prepare papers, hold seminars and summer schools in order to train MIIGAiK students.

4. The technological field

To develop software for three-dimensional modeling of gravitational fields of small celestial bodies.