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Astron. Astrophys. 321, 177-188 (1997)

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6. Summary and discussion

We have investigated the thermal and mechanical equilibria of magnetic loops on rapidly-rotating stars. These loops are assumed to be slender and are embedded in a potential arcade located on the stellar equator. We have analysed the effects of two different heating functions: one that is proportional to the density and one that falls off exponentially with height. While the total energy deposited in the loop depends on the form of heating function used, the summit temperature and pressure are largely unaffected, as are the radiative losses. As a result, it appears impossible to tell from present observations which heating mechanism is operating. The main factor governing the thermal properties of the loop solution is in fact the conductive flux at the base, which we impose as a boundary condition. The shape of the loop on the other hand depends mainly on the ratio of the internal and external plasma pressures, which again is imposed at the base of the loop.

Four different classes of solution have emerged from this study. These may be over- or under-pressured compared to their environment and either hotter or cooler. Looking at each case in turn we find the following.

  • Over-pressured, hot loops can be produced if the base conductive flux or the heating is enhanced. As we look at progressively taller equilibria for these loops, the foot points first separate, reaching the maximum separation of the arcade, then come together again. These loops are generally over-dense, but under-dense cases can be found if a large base conductive flux is imposed, giving a large temperature gradient.
  • Over-pressured, cool loops which are always over-dense can be produced if the base conductive flux or the heating is suppressed and the internal pressure is enhanced. The behaviour of the foot point separation is essentially as described above, i.e. the foot points first move apart then come together as the height is increased. Above the co-rotation radius, over-dense loops tend to be less tall than their under-pressured counterparts.
  • Under-pressured, cool loops can be produced if the base conductive flux is suppressed. The higher conductive flux solutions tend to be over-dense, with foot points that separate with increasing loop height. The effect of very rapid rotation making the loop summit outwardly-buoyant can, however, cause the foot points to come together again. Under-dense solutions can be produced with lower conductive flux values. For these cases, the foot point separation increases with increasing loop height.
  • Under-pressured, hot loops can arise when the base pressure in the loop is lower than in the arcade, though some solutions with low initial loop pressures show a pressure cross-over where the loop pressure will rise above the arcade pressure. The foot points of truly underpressured loops move apart as the loops rise; for loops with a pressure cross-over, the foot point separation tends to widen.

It is clearly possible to produce loops with a wide range of temperatures and structures, simply by adjusting their base values of heating, conductive flux or pressure. Since we have no reason to expect that a rapidly-rotating star will have a uniform surface, we might expect that the corona will be also be structured, both spatially and thermally. The results of the Mt. Wilson study (Baliunas et al. 1995) and Doppler imaging results (e.g. Unruh et al. 1995) show that many rapidly-rotating stars do indeed have starspots, and so have a highly-structured magnetic field at the surface. The discovery of stellar prominences (Collier Cameron & Robinson 1989) also demonstrates that the large-scale coronae of rapidly-rotating stars can also be highly-structured, with closed magnetic loops extending out to several stellar radii.

While rotation can drastically alter the pressure stratification within a loop (which may be crucial for the formation of stellar prominences at the co-rotation radius), its influence on the loop shape is relatively modest if all other parameters are kept constant. Where rotation may have a very important effect, however, is in determining those very base conditions that characterise the thermal and mechanical properties of the loop. If we assume that the base magnetic field strength varies with rotation [FORMULA] as [FORMULA], then equipartition requires that [FORMULA]. On the basis of this alone we would expect that the nature of coronal loops should vary with rotation rate. The other critical factor in determining the loop behaviour is the conductive flux. We can use the observed values of the differential emission measure (DEM) for a sample of stars of known rotation rate to determine empirically how the conductive flux varies with rotation rate. We do this by looking at the DEM for the C IV line which is formed in a fairly narrow temperature range. At a constant temperature, the conductive flux is proportional to the square of the density (or pressure) divided by the DEM. We find from observations that DEM [FORMULA] where x lies between 1 and 1.5, implying that the conductive flux [FORMULA] varies with rotation as [FORMULA].

Using these scalings we have compared our models with the observed temperatures and X-ray fluxes of stars of differing rotation rates. We tried three prescriptions for the magnetic field variation: [FORMULA]. The best fit to the observed temperatures is obtained with [FORMULA], which also gives a reasonable fit to the X-ray flux at least for the intermediate rotators. Whereas the radiative losses, loop summit temperature and summit pressure are very similar for all three heating parameterisations, the total heat deposited in the loop depends somewhat on the choice of the heating function. The heat deposited is not a measurable quantity, though, so that we cannot use this model to differentiate between the different heating mechanisms.

While our models can explain the variation with rotation rate in the temperature and X-ray flux from stellar coronae at low rotation rates, simply by allowing the magnetic field strength to scale with rotation, it is clear that some other mechanism is needed to explain the more rapid rotators. The saturation of the X-ray flux at high rotation rates has been known for some time and is often attributed to a change in behaviour of the dynamo (Vilhu & Rucinski 1983; Hempelmann et al. 1995).

There are two effects that could result in the saturation effect that is observed in the data. Firstly, equipartition (i.e.  [FORMULA]) might not hold for the faster rotators, if e.g. the gas pressure cannot continue to increase as the magnetic field strength (or the packing of the loops) increases. Secondly, the dynamo could change configuration, so that the scaling ([FORMULA]) of the magnetic field strength as a function of rotation rate changes. From our models, we see that reducing the value of q, which corresponds to a weaker dependence of the magnetic field strength on rotation rate, gives lower values for the X-ray flux. We have not pursued a model with variable q because of the lack of data on the C IV DEM at high rotation rates.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998