\documentstyle{article}
 \title{On the origin of the Equinox Non-Precession Motion in Fitting
       dynamical with fundamental reference frames.}
                      \author{ D.P. Duma\\
(Main astronomical Observatory of National Academy of Sciences of\\
                         Ukraine)}
\date{}


\begin{document}
\maketitle


       Using an angle measuring instrument, The geometrical model of
the principal circles and points on the sphere can be constructed by
$\alpha_k$, $\delta_k$ for the Sun obtained by the differential method in the
system of the catalogue K together with $\alpha_e$, $\delta_e$ calculated by the
adopted motion theories. This model is used then to derive equations
containing unknown corrections to the catalogue zero-points and
corrections to the Earth's orbital elements. We obtain the equations
of the form:
\begin{equation}
 [ \alpha_k - \alpha_e , \delta_k -\delta_e ] = [\Delta A, \Delta \delta_0,
\Delta L_0, \Delta_{\mu}, 2e\Delta \pi, 2\Delta e, \epsilon_1 - \epsilon_e ] M_1                   \label{eq:eq1}
\end{equation}

In the equation (\ref{eq:eq1}), $\Delta A$ and $\Delta \delta_0 $ denote the corrections to $\alpha_k$ and $\delta_k$ for reducing them
to the dynamical zero-points. For comparison we consider the equation
system generally used in practice for reduction of observations of the
Sun:
\begin{equation}
 [ \alpha_k - \alpha_e , \delta_k -\delta_e ] = [\Delta A, \Delta \delta_0,
\Delta L_0^{'}, \Delta_{\mu}^{'}, 2e^{'}\Delta \pi^{'}, 2\Delta e^{'}, \Delta\epsilon_1 ] M_2                   \label{eq:eq2}
\end{equation}

$\Delta \alpha_0$   in the equation \ref{eq:eq2} is also generally called as a correction to the catalog right
ascensions. We show that some elements of the matrix $M_2$   differ from
the corresponding elements of $M_1$  , therefore the primed corrections in
(\ref{eq:eq1}) and unprimed in (\ref{eq:eq2}) are distinct by their geometrical essence.
       Comparing (\ref{eq:eq1}) with (\ref{eq:eq2}) we have
\begin{equation}
\Delta \alpha_0  = \Delta A + (\lambda _1 - \lambda_e) cos\epsilon
\label{eq:eq3}
\end{equation}


In (\ref{eq:eq3}), the difference of longitudes reckoned from the catalogue and
dynamical equators can be represented through the corrections to the
Earth's orbital elements by the formulae:

\begin{equation}
\lambda_1 - \lambda_e = (1 + 2ecosM)\Delta L_0 + (1+2ecosM)(t_i-T_0)\Delta\mu \\
  - cosM.2e\Delta\pi   + (2+5ecosM)sinM.\Delta e  \label{eq:eq4}
\end{equation}

Subtituting in (\ref{eq:eq2}) the initial element epoch $T_0$ by the mean moment of the observation sets $t_0 = 1/N.\sum_{i=1}^N t_i$ (a procedure followed in reducing
real observations), we obtain solutions for $\Delta\alpha_0$ and
$\Delta\delta_0$. Comparing the
estimates of  $\Delta\alpha_0$ obtained for different observation sets of the Sun we can
find a relationship of the form

\begin{equation}
(\Delta\alpha_0)_{t_0} = (\Delta \alpha_0)_{T_0} +  \Delta \dot\alpha (t_0 -T_0)
\label{eq:eq5}
\end{equation}

where  $\Delta\dot\alpha$ is the so-called non-precessional equinox motion per century.
    The equinox motion $\Delta\dot\alpha\simeq 1.3"$  is truly known in practice according to 
w. Fricke.
    With this in mind, the inference is made that the non-precessional
equinox motion is a result of the reduction procedure of observations
of the Sun and planets used in practice together with the imperfect
Earth's motion theory proposed by Newcomb as early as in 1895 and
applied above 70 years for calculating $\alpha_e$ and$\delta_e$     
   The main result are as follows. A more perfect geometry model is
constructed for fitting the stellar and planetary reference frames. The
equation system is obtained for calculating the zero-point corrections
of the fundamental catalogue. An explanation of the non-precessional
equinox motion is proposed. The discrepancy is found between the
geometrical interpretation of values determined by the traditional
method and the method suggested in this paper. The results of this
investigation are of fundamental importance for solving the problem on
fitting the stellar reference frame with planetary one using the
ground-based and space observations of the Solar system bodies.   

\end{document}
