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             \author[1]{Milutin  Marjanov}

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\date{\small \em Received: 10 December 2013 Accepted: 2 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


Concept of homogeneity (or inhomogeneity) of the gravitational field is a rather specific one. Namely, according to the Newton`s law of gravitation, either intensity or direction, or each of these two characteristics of the gravitational force acting on the body, depend on its position in the gravitational field. So, in fact, for the body as a whole, the gravitational field is always an inhomogeneous one. Such a classification makes sense only if it is restricted to the active part of the gravitational field, that is, on the part occupied by the body.

\end{abstract}


\keywords{homogeneous and inhomogeneous gravitational field newton?s gravitational force gravitational moment.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
oncept of homogeneity (or inhomogeneity) of the gravitational field is a rather specific one. Namely, according to the Newton`s law of gravitation, either intensity or direction, or each of these two characteristics of the gravitational force acting on the body, depend on its position in the gravitational field. So, in fact, for the body as a whole, the gravitational field is always an inhomogeneous one. Such a classification makes sense only if it is restricted to the active part of the gravitational field, that is, on the part occupied by the body. 
\section[{II. Homogeneous and Inhomogeneous Gravitational Field}]{II. Homogeneous and Inhomogeneous Gravitational Field}\par
Consider a body of mass m moving in the gravitational field of the dominant gravitational center of mass m*. Assume that m/m*??1, so the gravitational field is a stationary one. Gravitational noice, as well as the gravitational anomalies, are excluded. The motion of the body is composed: while moving in its orbit, it revolves about its principal central axis of inertia \hyperref[b0]{(1)}, which is perpendicular to the orbital plane. The mass center C of the body is chosen to be at the origin of two moving frames of reference xCy and ?C?. The first one is related to the geometry of the orbit, Cx being oriented toward the gravitational center, that is ? the positions of the particles (?,?) within the body, nor of the angle of relative rotation ?. All these forces have the same direction, that is, they are parallel to the coordinate axis Cx and their sum, the "weight" of the body, coincides with that line, regardless of the relative position of the body in the frame of reference xCy. In a homogeneous gravitational field the resultant of the ellementary gravitational forces always passes through the mass center of the body and the gravitational moment does not exist. In fact, the mass center of the body in a homogeneous gravitational field represents the center of gravity, as concieved by Archimedes some 2,5 centuries B.C. (Fig.  {\ref 2}). Generally, this resultant doesn`t pass through the mass center of the body, so it has to produce the gravitational moment (Fig.  {\ref 3}).\par
Within the first order inhomogeneity case, the gravitational force is the same as in the homogeneous gravitational field2 * R m m G F ? =\par
Where G is the gravitational constant and the gravitational moment for the described motion is equal? ? ? = 2 sin R ) I I ( m G 2 3 M 3 3 2 * C\par
, Where I 2 and I 3 are the medium and the minimum principal moments of inertia for the mass center of the body. Obviously, the gravitational moment is a harmonic function of the double angle of relative rotation, with the amplitude depending on the mass of the gravitational center, on the distance between the body and this center and finally, on the mass distribution in the body.\par
When the small body goes around the large body in a closed orbit, its orbital and rotational motions gradually become resonant just because of the gravitational torque existence (/3/,/4/). For example the Moon circulates around Earth and rotates around its axis in the 1/1 resonance\par
The first characteristic of inhomogeneity of the gravitational field is the existence of the gravitational moment, in the general case. The second one is the absence of the center of gravity of the body. 
\section[{III.}]{III.}\par
Body without Center of Gravity Authors of many of textbooks are not quite precise about that point. Having, probably, in mind the existence of the gravitational moment in such a field, they often claim implicitly, or even explicitly, that in an inhomogeneous gravitational field the center of gravity doesn't coincide with the mass center of the body. This is definitely wrong, of course, because such an assertion may lead the reader to the false Fig.  {\ref 4} : Oscillation of the gravitational force around the gravitational center conclusion that there exists something like a "moving center of the gravity" in the rotating body exposed to the inhomogeneous gravitational field. There is no such a point in, or in the vicinity of the body, satisfying the definition of the center of gravity, if the body is exposed to the field of the convergent gravitational forces. The "weight" vector of the body has to pass through the point toward which converge all its components and that is the gravitational center. The relative rotation of the body produces tilting of this vector about this center (Fig.  {\ref 4}).\par
The angle between the gravitational force and the direction CC* is equal  
\section[{Conclusion}]{Conclusion}\par
We have stressed the distinction between homogeneous and inhomogeneous gravitational fields. A homogeneous gravitational field is marked by the existence of the center of gravity and the absence of the gravitational moment acting on the body. On the other hand, in an inhomogeneous gravitational field the body has no center of gravity and the gravitational moment, generally, exist. Concerning the calculus of the \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
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\noindent\textbf{23}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 2 :Fig. 3 :}\end{figure}
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\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_3}}\end{figure}
 			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Misner et al. (ed.) ()]{b1}\label{b1} 	 		\textit{},  		 			C W Misner 		,  		 			K S Thorne 		,  		 			J A Wheeler 		.  		Gravitation, W.H. Freeman and Comp (ed.)  		1973. San Francisco.  	 
\bibitem[Marjanov ()]{b2}\label{b2} 	 		‘Gravitational Resonances’.  		 			M Marjanov 		.  	 	 		\textit{Bull. Astron}  		1997. 156 p. .  	 
\bibitem[Hawking and Israel ()]{b0}\label{b0} 	 		 			S Hawking 		,  		 			W Israel 		.  		\textit{300 Years of Gravitation},  				1989. Cambridge University Press.  	 
\bibitem[Marjanov]{b3}\label{b3} 	 		\textit{One Solution of the Newton's Three Body Problem},  		 			M Marjanov 		.  		 \url{www.milutinmarjanov.com}  		 	 
\end{bibitemlist}
 			 		 	 
\end{document}
