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Astron. Astrophys. 321, 151-158 (1997)

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3. Results and discussion

We begin by considering the case of equal radii, [FORMULA], say, and vary the separation of the centres. We take a fixed value, [FORMULA], at which a dynamo is excited over a wide range of radii, [FORMULA]. Calculations are usually started from an arbitrary configuration containing a mixture of [FORMULA] and [FORMULA] components, with [FORMULA]. Results are outlined in Table 1. For [FORMULA], the dynamo is not excited at [FORMULA]. This sequence of computations exhibits several interesting features. For [FORMULA], the solutions are steady, of type E, and have [FORMULA]. M increases as [FORMULA] decreases and the relative separation of the centres increases. For [FORMULA], the solutions are steady, of type O, and [FORMULA]. The behaviour for larger [FORMULA] (ie smaller separation of centres) is consistent with the limit [FORMULA], as discussed in Sect. 2. For [FORMULA], both O and E type solutions coexist. Here [FORMULA], so the overall solution is of mixed parity, [FORMULA], and it is oscillatory. For values of [FORMULA] a little smaller than 0.5, the O-type solutions disappear, and solutions are again steady. Between [FORMULA] and 0.410 there is a further bifurcation, and the stable solution becomes oscillatory, with constant parity [FORMULA]. In Fig. 2 we show some details of the temporal behaviour of the [FORMULA] and [FORMULA] solutions, and Fig. 3a,b gives field vectors projected on to equatorial and azimuthal planes when [FORMULA]. In Fig. 4a,b for the [FORMULA] calculations we give contours of radial and absolute field at the surface of the computational sphere, and in Fig. 4c,d the corresponding quantities at the surface of one of the dynamo-active component spheres.


[TABLE]

Table 1. Summary of results for equal stellar radii, [FORMULA]. A - in the 5th column indicates that [FORMULA] fields were not present. The column headed `type' distinguishes steady and oscillating solutions. An entry of two bracketed values indicates the range of variation of an oscillatory quantity. [FORMULA] after an entry indicates small oscillations near the tabulated value.

[FIGURE] Fig. 2a-c. a Behaviour of P (solid), M (long-dashed), [FORMULA] (medium- dashed) and [FORMULA] (short-dashed) for the [FORMULA], [FORMULA] solution. b Behaviour of [FORMULA] (solid), [FORMULA] (long-dashed), [FORMULA] (medium-dashed) and [FORMULA] (short-dashed) for the same solution as a. c   As b for [FORMULA], [FORMULA].


[FIGURE] Fig. 3a and b. [FORMULA], [FORMULA]. a Field vectors in the equatorial plane, projected on to that plane. The line of centres passes horizontally through the centre of the figure. [FORMULA] is to the right and [FORMULA] increases anticlockwise. The enclosing circle is the projection of the computational volume. b Field vectors in the plane [FORMULA], projected on to that plane. The rotation axis passes vertically through the centre of the figure.
[FIGURE] Fig. 4a-d. [FORMULA], [FORMULA]. a Contours of radial field strength on the surface of the computational sphere. b Contours of absolute field strength on the surface of the computational sphere. [FORMULA] runs horizontally from [FORMULA] to [FORMULA], [FORMULA] vertically from 0 (top) to [FORMULA] (bottom). c Contours of radial field strength on the surface of the primary component. d Contours of absolute field strength on the surface of the primary component. In c and d, if [FORMULA], [FORMULA] are spherical coordinates measured with respect to the spin axis of the primary, then [FORMULA] runs from [FORMULA] (top) to [FORMULA], and [FORMULA] from 0 to [FORMULA] horizontally, where [FORMULA] corresponds to the direction of the centre of the secondary

In Table 2 we present several other calculations for equal radii, but with different [FORMULA] values. No significant changes in behaviour are apparent from those with [FORMULA], except that when [FORMULA], the [FORMULA] solution is steady and that with [FORMULA] is only marginally excited.


[TABLE]

Table 2. Summary of results for equal stellar radii, other values of [FORMULA].


Table 2 gives results from computations for detached configurations of unequal radii. As anticipated, the solutions are now of type W, and so, necessarily, [FORMULA]. Usually the solutions are of `pure' parity, [FORMULA], but we did discover some stable `mixed parity' solutions. Solutions can be either oscillatory or steady. In some cases, very long-lived transient behaviour is observed, over as long as 50 or more global diffusion times. In the cases with [FORMULA] in Table 2, the fields appear to be settling to a [FORMULA] configuration when, after 10-15 diffusion times a relatively sudden change to [FORMULA] occurred, accompanied by a decrease in M from near unity to the tabulated values. We show details of the evolution of parity and M with time for this case in Fig. 5a, and of the energies for the case ([FORMULA], [FORMULA], d) = (0.4874, 0.30, 1.05), [FORMULA] in Fig. 5b. As anticipated in Sect. 2, solutions listed in Table 2 with [FORMULA] values close to those of the O-type solutions in Table 1 do have the majority of the energy in the odd m field modes. In Fig. 6a,b we show field contours over the surface of the computational sphere, and the surface of the primary component, for the solution shown in Fig. 5b. The last entry of Table 2 is noteworthy. Although the parameters are only slightly different from those of the immediately preceding entry, its behaviour is quite different in that the fields are not then of pure parity, [FORMULA].

[FIGURE] Fig. 5a-b. a Evolution of P, M, [FORMULA], [FORMULA] when [FORMULA], [FORMULA], as Fig. 3a). Temporal behaviour of modal energies [FORMULA], [FORMULA], [FORMULA], [FORMULA], for [FORMULA], [FORMULA] calculation, as Fig. 3b)
[FIGURE] Fig. 6a and b. [FORMULA], [FORMULA] b Contours of absolute field strength on the surface of the computational sphere. c Contours of radial field strength on the surface of the primary component. d Contours of absolute field strength on the surface of the primary component. Orientation is as described for Fig. 4
[FIGURE] Fig. 6c and d. continued

In Table 3 there are results for contact configurations of unequal radii. These solutions are all of type W and are steady with [FORMULA]..


[TABLE]

Table 3. Summary of results for detached configurations with unequal stellar radii. A [FORMULA] indicates computations with the origin of coordinates at the centre of mass.



[TABLE]

Table 4. Summary of results for contact configurations with unequal stellar radii. A [FORMULA] indicates computations with the origin of coordinates at the centre of mass.


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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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