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*Astron. Astrophys. 350, 434-446 (1999)*
## Appendix A: propagation of dispersions with time in the epicycle theory
Let us consider the vector =
(,,,,,)
containing the position and velocity of a star at a given time
*t*. We are interested in determining the propagation of the
observational dispersions over time, in the epicycle approximation
frame (Eqs. 1, and their derivatives). If we define the vector
= (,,,,,)
which contains the initial position and velocity of our star, then the
initial dispersions can be propagated with time as:
where is the observed dispersion
in the variable , and
the dispersion at *t*. The
dependence of variables on the
initial values is given by Eqs. 2.
If the coefficients of epicycle equations are included in a vector
= (,,,,,)
then the matrix in Eq. A1 can be
determined as the matrix product
= ,
where the matrix and
(). By partially differentiating
Eqs. 1 we can determine the
matrix (Table A1),
whereas matrix is determined by
just differentiating Eqs. 2 (Table A2). Now, from matrices
and
we can determine the
matrix , given in
Table A3. From matrix ,
and by means of the relationship A1, the propagated dispersions
can be expressed as:
From these equations we observe that all the dispersions in
variables oscillate around a
constant value, except , whose
averaged value increases with time. For small values of *t*
Eqs. A2 can be approximated by:
**Table A1.** Elements of the matrix (see text)
**Table A2.** Elements of the matrix (see text)
**Table A3.** Elements of the matrix (see text)
© European Southern Observatory (ESO) 1999
Online publication: October 4, 1999
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