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Astron. Astrophys. 350, 434-446 (1999)

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Appendix A: propagation of dispersions with time in the epicycle theory

Let us consider the vector [FORMULA] = ([FORMULA],[FORMULA],[FORMULA],[FORMULA],[FORMULA],[FORMULA]) 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 [FORMULA] = ([FORMULA],[FORMULA],[FORMULA],[FORMULA],[FORMULA],[FORMULA]) which contains the initial position and velocity of our star, then the initial dispersions can be propagated with time as:

[EQUATION]

where [FORMULA] is the observed dispersion in the variable [FORMULA], and [FORMULA] the dispersion at t. The dependence of variables [FORMULA] on the initial values [FORMULA] is given by Eqs. 2. If the coefficients of epicycle equations are included in a vector [FORMULA] = ([FORMULA],[FORMULA],[FORMULA],[FORMULA],[FORMULA],[FORMULA]) then the matrix [FORMULA] in Eq. A1 can be determined as the matrix product [FORMULA] = [FORMULA], where the matrix [FORMULA] and [FORMULA] ([FORMULA]). By partially differentiating Eqs. 1 we can determine the matrix [FORMULA] (Table A1), whereas matrix [FORMULA] is determined by just differentiating Eqs. 2 (Table A2). Now, from matrices [FORMULA] and [FORMULA] we can determine the matrix [FORMULA], given in Table A3. From matrix [FORMULA], and by means of the relationship A1, the propagated dispersions can be expressed as:

[EQUATION]

From these equations we observe that all the dispersions in variables [FORMULA] oscillate around a constant value, except [FORMULA], whose averaged value increases with time. For small values of t Eqs. A2 can be approximated by:

[EQUATION]


[TABLE]

Table A1. Elements of the matrix [FORMULA] (see text)



[TABLE]

Table A2. Elements of the matrix [FORMULA] (see text)



[TABLE]

Table A3. Elements of the matrix [FORMULA] (see text)


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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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