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We consider the problem of determining the center of mass of an unknown gravitational body, using the disturbances in the motion of observed celestial bodies. In this paper an universal approach to obtain the approximate and stable estimate of problem solution is suggested. This approach can be used in other fields of Science. For example, it can be applied for investigation of interactions between fields of forces and elementary particles using known trajectories of elementary particles motions.

In 1843-1845 famous astronomers and mathematicians Urbain Jean Le Verrier (1811-1877) and John Couch Adams (1819-1892) independently of one another performed the mathematical research and came to the conclusion that the Solar system includes a celestial body (at least one) which has not observed earlier.

In fact not only the existence of a previously unknown planet has been proven, but also its orbit has been determined with an accuracy, which was sufficient for its detection and surveillance. The planet Neptune has been discovered as result. Mathematically these problems belong to the category of inverse problems of mathematical physics, i.e., to ill-posed problems. The solution of this problem was executed by the method of least squares using some hypothesizes. After discovering Neptune, Le Verrier started the recalculation of theory motion of Uranus by taking into account the motion of Neptune. After finishing his investigation, Le Verrier was able to achieve results with high accuracy, which unfortunately still disagreed with results obtained by observation. This difference was not due to an error in theory or observation [

Much later it was discovered a natural property of inverse problems their instability.

Of course, our knowledge of the Solar system has not been and will never be final and the level of our knowledge is entirely determined by the level of theoretical and observational studies. However, a theoretical analysis of the constructed motion of large (and primarily external) planets indicates, that there are yet unexplained discrepancies between theory and observation. Despite the fact that the theoretical parameters of motion were refined with results made from observation, which were made over a long period of time. For example, there are latitudinal variations in the motion of Uranus and Neptune and the deviation in the movement perihelion of Halley’s comet, that cannot be explained by gravitational forces of known solar system bodies. These circumstances have led to the fact that in the 60s of the last century a hypothesis for the existence of a tenth planet emerged. This 10th planet should have a mass equal to the mass of Jupiter, with an approximate distance to the sun of 60 AU and an orbital tilt of

Analysis of solution methods of Le Verrier and Adams shown these methods did not take into account the inaccuracy of mathematical model of planets motion. The success of their investigations was guaranteed with help of right hypothesis about tilt of unknown planet orbit to plane of the ecliptic and orbit eccentricity.

This fact is explained failure of big numbers of investigations after Le Verrier for searches of planet Pluto.

Thus the development of stable methods of approximate solutions of the inverse problem astrodynamics in more general statement remains relevant.

We consider n interacting masses moving under the forces of mutual attraction in an inertial coordinate system. Masses

where G is the gravitational constant.

The mass

where

Let us make the transition of the variables

It is assumed that among n gravitational masses the location of only mass

Equation (3) takes the form

where

In terms of projections on the axis of the inertial coordinate system the Equation (4) can be written in the form:

where

Let us integrate Equations (5)-(7) twice from

where

Each equation of the system Equation (8) can be presented in the form

where

Equations (9) are known as Volterra integral equations of the first kind with respect to the unknown functions

By finding the solutions of the Equations (9)

Performing similar calculations and solving equations of the type Equation (9) for the mass

As is easily seen,

Solution of the Equation (9) in the physical sense must also belong to

In the Equations (5)-(7) of motion the coefficients

Introduce into consideration the following notations

where

The Inequalities (10) define a closed region of

where

The Equation (9) can be written in the form

where the operator

The operator

We denote by

Suppose instead

The approximate value of

Let us estimate the deviation of

where

Since real processes can be described by mathematical methods only approximately. It is assumed that the exact operator

In this case it is possible to use the algorithm for solving the inverse problem with approximate operator

However, the assumption of linearity of the exact operator

Then the set of possible solutions

It is easy to show that if the operator

For solving ill-posed problem Equation (11) we use the Tikhonov regularization method with the stabilizing functional

where

Thus, the problem of finding an approximate solution of Equation (11) reduces to the solution of the extreme problem:

It should be noted that there is no way to determine the size of

Therefore, the approximate solution of inverse problems of measurement are not of interest for practical use due to instability of the solution.

The way out of this impasse exists, if by the investigation of inverse problems of measurement restrict only some estimates of exact solutions.

The set of possible solutions

Let us considered the following extreme problem

The regularization parameter

To obtain useful information on the exact solution of the inverse problem the use of the following hypothesis (main hypothesis) is suggested: the inequality is valid

where the function

If the exact operators

Evaluation of the exact solution

The estimate

If

Note that the preparation of estimates does not use properties of the exact operators

We give sufficient conditions for the existence of an element

Theorem1. Let

tional

In order to study the influence of the process parameters on the estimation of the exact solution, we consider the union of the sets

The set

Let

The regularization parameter

Since

is valid.

Evaluation of the exact solution of

To study the influence of the parameters p on the estimation of the exact solution of the inverse problem it is necessary to have a possibility to select an operator

implies the inequality

for any possible

Subsequently, the operator

If the operator

holds.

The problem (26) has a solution for every

In this case the following inequality holds for any

Evaluation of the exact solution

Theorem 2. Special minimal operator

Proof. Let

For any

The function

where C is a real symmetric matrix

Coefficients of the matrix C are given by:

Since

A necessary and sufficient conditions for strong convexity of

for any

Quadratic form (28) is positive as

Therefore,

Similarly, [

for any

Suppose that among the operators

then the inequality

for any possible

Subsequently, the operator

operator” in the sense of satisfying Inequality (30).

If the operator

extreme problem: find an element

holds.

The problem (31) has a solution for any

Thus obviously the following inequality is valid:

Theorem 3. Special maximal operator

for any

Proof. The proof is similar to Theorem 2. □

In this paper we proposed an algorithm for finding the coordinates of an unknown gravitational mass as a result of astronomical observations. This problem was solved first by Urbain Jean Le Verrier and John Couch Adams. Here a more universal approach was suggested. The proposed hypothesis allows us to exclude the error of the approximate operator from the calculations. Also conditions for the existence of an approximate solution were obtained and several non-standard formulations of inverse problems were considered. Suggested approach can be used in other fields of Science. For example, it can be applied for investigation of interactions between fields of forces and elementary particles by help of known trajectories of elementary particles motions.

YuriMenshikov, (2015) Inverse Problem of Astrodynamics. World Journal of Mechanics,05,249-256. doi: 10.4236/wjm.2015.512023