**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.