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Astron. Astrophys. 341, 567-573 (1999)

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5. The streamlines for constant Jacobi energy

In the chosen coordinate system the general equation of the streamlines is:

[EQUATION]

Substituting for [FORMULA] and [FORMULA] from Eqs. (25) and (26) we find:

[EQUATION]

Using the identity:

[EQUATION]

we can rewrite Eq. (30) as:

[EQUATION]

which integrates to give:

[EQUATION]

or, for the choice of A given in Eq. (28):

[EQUATION]

Depending on the choice of the "streamline constant" in Eq. (34) the streamlines are open or closed. The limiting streamline separating the two groups is given by:

[EQUATION]

The properties of this limiting streamline are as follows. The streamline begins at a stagnation point [FORMULA] on the equator. It then moves inwards, at first at almost constant [FORMULA], and crosses the `top' of the star at [FORMULA]. It then continues over to the other side, making this excursion as far as [FORMULA]. After passing through this extremity, it returns over the top of the star again, this time at [FORMULA], finally returning to the equator again at a second stagnation point located at [FORMULA].

The very large loop which we have described above together with the remaining equatorial segment between [FORMULA] and [FORMULA], encircles all of the closed streamlines. Recalling that the sink and source are located at [FORMULA] and [FORMULA] respectively, we see that the system of closed streamlines covers almost the whole surface of the star. The `open' streamlines connecting the source and sink are confined to a C-shaped narrow channel around the equator so that the material flowing from source to sink can only proceed along that side of the equator on which no stagnation points occur. The maximum width of the channel amounts to only [FORMULA].

This particular flow topology is a direct consequence of the action of the Coriolis forces. To see this, let us simply omit the Coriolis term in Eq. (11) (i.e. the third term on the L.H.S.) and repeat the analysis. We then find instead of Eqs. (25) and (26) the solution:

[EQUATION]

corresponding to a complete system of open streamlines. One can hardly imagine a more dramatic change in the flow topology.

An immediate consequence of the change in flow topology caused by the Coriolis forces is that most points on the stellar surface are not accessible to a streamline coming from the source or going to the sink. Hence it is not valid to argue that the Jacobi energy must be constant over the surface of a contact binary on the grounds that all streamlines can be traced back to a point where they all come together. This argumentation was used for contact binaries of the `reversing layer' type in K 97.

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998
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