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Astron. Astrophys. 334, 381-387 (1998)

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5. A semi-analytical relation between [FORMULA] and [FORMULA]

Our aim is to find a semi-analytical relation [FORMULA] for the intervals [FORMULA] and [FORMULA].
The first step is the determination of a function [FORMULA] for a fixed value of [FORMULA]. We solve Eq. (7) for [FORMULA] and [FORMULA] and by the least square method we find the function [FORMULA]:

[EQUATION]

with:

[EQUATION]

The value of the correlation coefficient between this function and the numerical integration is [FORMULA]. With this method we find [FORMULA] for several values of [FORMULA] inside the interval [FORMULA]. We can write the dimensionless collapse time as the product:

[EQUATION]

where the function [FORMULA] can be written as

[EQUATION]

and, for [FORMULA] and [FORMULA], K is constant: [FORMULA]. So:

[EQUATION]

A [FORMULA] test performed between the values obtained for [FORMULA] from the numerical integration and the values obtained from Eq. (7) gives the result: [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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