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

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5. Importance of the rotational velocity

We can ask whether our model can predict coronal parameters for a range of rotation velocities that varies between the solar value and the much higher rotation rates that are observed on young and active stars. Jordan & Montesinos (1991) and Montesinos & Jordan (1993) have looked at the coronal temperatures and emission measures as a function of rotation rate. They find that the coronal temperature is proportional to the rotation rate, so that [FORMULA]. They also found a relationship between temperature, gravity and emission measure (EM), [FORMULA].

Depending on the boundary conditions that we choose, our model agrees reasonably well with these scaling laws. In the following, we assume that the magnetic field strength scales with rotation rate, so [FORMULA]. The most common choice is to set [FORMULA], but some observational results seem to indicate that the slope is somewhat flatter and q could be as low as 0.5 (Montesinos & Jordan 1993; Baliunas et al. 1996). Assuming equipartition, the gas pressure will then be proportional to the square of the rotation rate, [FORMULA]. If we set the loop foot points to a fixed temperature, the conductive flux is proportional to the square of the density (or pressure) divided by the differential emission measure (DEM).

Fig. 9 shows the DEM of the C IV line as a function of rotation rate for single F, G and K stars. From this we deduce [FORMULA], with x about 1.5. The DEM in C IV is produced in a relatively narrow temperature range. We can hence use it to determine the scaling for the conductive flux at the base, [FORMULA].

[FIGURE] Fig. 9. The logarithm of the differential emission measure of C IV. The triangles are for data from Jordan et al. (1987), the squares are for data from Giampapa et al. (1985) and the diamonds are for data taken from Ayres et al. (1995). The DEM is measured in units of cm-5, and the data from Ayres et al. (1995) and Giampapa et al. have been scaled by using measurements of stars that were also measured by Jordan et al. The star showing a very large value for the DEM at [FORMULA] is 110 Her; its rotation period has been deduced using its [FORMULA] value and could be higher if it was viewed at a low inclination (Jordan & Montesinos 1991). The dashed line has a slope of -1.5 and corresponds to the linear least-square fit to the data excluding 110 Her. For comparison, the dotted line has a slope of -1.

5.1. Results

We fixed the boundary conditions close to values observed on the Sun and then increased the rotation rate until we reached at least 50 times the solar value. For the first runs we assumed that [FORMULA], so that [FORMULA], [FORMULA] and [FORMULA]. The value for the initial temperature was kept fixed at [FORMULA]  K. The advantage of setting the boundary temperature to [FORMULA]  K is that we can use the C IV emission that is formed at this temperature to set the conductive flux value. The disadvantage is that one does not necessarily know whether the loop solutions are connected to the chromosphere (van den Oord & Zuccarello 1996). Also, one tends to underestimate the radiative losses and the total heat input required when starting with too high a value for the temperature at the loop foot point. We have compared the loop summit temperatures and pressures for loops with boundary temperatures of [FORMULA]  K and [FORMULA]  K (below which the gas becomes optically thick). We found no difference between the behaviour of the different loop solutions when we used the same scaling laws, except for a very small temperature and pressure offset.

For all runs, the loop length was fixed at [FORMULA] and the magnetic field fall-off was prescribed by setting [FORMULA]. At each rotation rate, the base heating was adjusted until the conductive flux fell below a threshold value.

It turned out that the increase in temperature is too steep if the magnetic field is proportional to the rotation rate. As some observational results indicate that the dependence of the magnetic field on the rotation rate is indeed flatter, we also tried scaling laws with [FORMULA] and [FORMULA]. For [FORMULA] we adopted [FORMULA], [FORMULA] and [FORMULA] ; this assumes that the DEM of C IV depends linearly on the rotation rate (see the dotted line in Fig  9). For [FORMULA] we adopted [FORMULA], [FORMULA] and [FORMULA]. This is clearly not realistic in terms of the behaviour of the C IV DEM as a function of rotation rate.

Fig. 10 shows a plot of the summit temperature versus rotation rate for all three scaling laws. Clearly, the summit temperature for [FORMULA] depicted by the solid line gives a much better fit than the linear dependence (dotted line) or a model where the magnetic field increases as the square root of the rotation rate (dashed line). The summit temperature is mainly a function of the conductive flux at the loop base. It is therefore not so much the behaviour of the magnetic flux as a function of rotation rate that determines the loop temperature, but the behaviour of the DEM. In fact, if we assume that the C IV DEM scales as [FORMULA] and [FORMULA], we have to use [FORMULA]. The resulting temperature scaling then very much resembles the dashed line in Fig. 10 that was obtained for [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 10. Plot of the temperature versus period. The dotted line is for a model with [FORMULA], [FORMULA] and [FORMULA]. The solid line is for [FORMULA], [FORMULA] and [FORMULA] and the dashed line for [FORMULA], [FORMULA] and [FORMULA]. The triangles are data from Jordan & Montesinos (1991); the square shows the coronal temperature and period for AB Dor.

Table 1 lists the base heating, the summit temperature and pressure, the total heat deposited and the radiative losses a loop for heating that is proportional to the gas density.


[TABLE]

Table 1. Initial and final values for the loops when the heating is proportional to the gas density. The magnetic field strength scales as [FORMULA].


Fig. 11 shows the X-ray flux as a function of period for a number of stars with [FORMULA] as listed in Hempelmann et al. (1995). The lines show the radiative losses in the loops. As in Fig. 10, the solid line is for [FORMULA], the dotted line for [FORMULA] and the dashed line for [FORMULA]. In general we find that our fits are slightly too flat for the slow rotators and fail to reproduce the flattening-off that is observed for the faster rotators. As the radiative loss is mainly a function of the square of the density, it is not very sensitive to any changes in the heating function. In Figs. 10 and 11, we have only plotted the temperature and the radiative losses for heating that is proportional to the density as the curves obtained with the other heating parameterisations coincide.

[FIGURE] Fig. 11. The X-ray flux as a function of rotation rate for a number of stars taken from Hempelmann et al. (1995). Also plotted is the radiative loss in a loop. The dotted line is for a model with [FORMULA], the dashed line for [FORMULA] and the solid line for [FORMULA]. The X-ray flux is in units of W m-2.
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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