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

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3. Details of the model

We do not have any observational evidence for the detailed nature of the ambient field on these rapidly-rotating stars, although Doppler images suggest that the foot points of large spot complexes may be separated by about [FORMULA]. The effect of the ambient field on the nature of the flux tube is principally through the effect of pressure balance. The most important feature of the external field therefore is the rate at which its magnetic pressure falls off with height. A dipole field, while appropriate for studying the largest-scale structures, is less suitable for studying the smaller-scale fields closer to the stellar surface where we expect most of the X-ray emission to originate. In order to study a range of spatial scales therefore, we choose to model the ambient field as a potential arcade with [FORMULA]. This allows us to change the scale-height of the external field simply by varying k. We have also investigated the effects of using a sheared arcade, but find that the results are qualitatively similar.

The presence of the flux tube will of course distort this arcade slightly as the flux tube pushes the arcade field lines out of the way. As we will show later, this distortion becomes significant when [FORMULA]. We ignore this effect when calculating the loop equilibrium, but calculate a posteriori the point at which it becomes important and reject solutions with large [FORMULA] values.

The tube can then be described by seven coupled first-order differential equations:


Eqs. (15) and (21) follow from the way we set up the coordinate system. Eqs. (17) and (19) are equivalent to Eqs. (13) and (14). Eq. (16) is the hydrostatic equation expressed in terms of s rather than y. Differentiating the equation describing the balance of the magnetic and gas pressure between the loop and the surrounding plasma, namely


yields (18). Eq. (20) is a rewrite of Eq. (10).

3.1. The surrounding plasma

To calculate the shape of the flux tube one needs to know the pressure structure of the plasma in which the tube is embedded. The usual approach is to assume that the plasma is isothermal, so that Eqs. (6) to (8) are valid.

If, however, the energy equation is used to calculate the temperature structure of the loop, the temperature increases very strongly within a short distance from the foot points. This means that, compared to the isothermal surrounding plasma, the pressure changes in the loop are dampened due to the inverse temperature dependence in Eq. (16). This can lead to unphysical effects, especially for loops that are higher than the co-rotation radius, when the density and pressure of the surrounding arcade become too large relative to the loop and can no longer be balanced by magnetic tension (see Sect.  3.2).

There is no a priori reason why the heating mechanism in the loops should be different from the heating mechanism of the corona. It is hence desirable to relax the isothermal assumption and to include the temperature variation of the surrounding plasma. This can be done fairly straightforwardly if the simplifying assumption is made that the loop shape does not vary considerably from the arcade shape. Then the path difference between the loop and the arcade can be neglected and the temperature, energy and pressure equations for the arcade can be solved in the same way as for the loop.

As the assumption that the surrounding plasma is isothermal has been dropped, Eqs. (15) to (21) are supplemented by three additional equations:


These correspond to Eqs. (16), (17) and (19), except that we can now use [FORMULA] and [FORMULA], as the shape of the arcade is fixed by our choice of a potential two-dimensional field for [FORMULA],


Note that because of the assumption of a potential field, the shape prescription will not be valid for large values of the
plasma [FORMULA].

3.2. The loop equilibrium

The loop shape is determined by the equilibrium of forces due to magnetic pressure, buoyancy and magnetic tension. The buoyancy term (which in the isothermal case is only a function of the pressure) is now a function of the temperature as well. Furthermore, it changes sign as the loop rises across the co-rotation radius. The effect on the loop shape can be seen by looking at the derivative of the magnetic field. From Eq. (18) we get


If [FORMULA] is small compared to unity we just recover the potential field solution. So the loop shape will only differ from the arcade if [FORMULA] becomes of the order of unity, or


In ideal MHD the deviation of the flux tube from the arcade is discontinuous at the boundary of the flux tube. In reality, we expect the flux tube to be surrounded by a current sheet that would contribute to the heating. The shape of a hot and over-pressured loop is shown as an example in Fig. 2a. Also shown are different arcade field lines. The dashed line shows the arcade field line that has the same gradient at its foot points as the loop. The lowest arcade is the arcade with the same foot point separation as the loop. Note that there is no maximum height for the arcades.

[FIGURE] Fig. 2a-d. The shape (a), temperature (b), pressure (c) and density (d) of a typical hot and under-dense loop. The solid lines are for the loop, the dashed lines for the surrounding arcade. The star is assumed to rotate 70 times faster than the Sun and to have a field strength of 0.3 T. The loop has a length of 4.6 stellar radii. The dotted lines in a show arcade shapes for different foot point separations. The dashed line is the arcade field line that was used to calculated the external parameters.

In most cases, the pressure difference in Eq. (29) will be much larger than the density difference, even if the loop rises above the co-rotation radius. This indicates that the plasma [FORMULA] has to be of the order of one in order for the loop shape to deviate. The equation also shows why, for a surrounding isothermal plasma, the rapid pressure increase of the arcade plasma relative to the loop plasma demands an increasing field strength with loop length beyond a critical point. As the external pressure increases rapidly, [FORMULA] becomes negative and starts to approach values of the order of unity. Eventually, the RHS of Eq. (28) becomes negative and [FORMULA] will have to increase.

3.3. The boundary values

To solve the set of Eqs. (15- 21, 23- 25), we need to specify ten boundary values. These are the internal and external pressures and temperatures, the external magnetic field, the conductive flux inside and outside the loop and the initial foot point position and gradient of the magnetic field line. We can specify most of the parameters at the foot points so that they agree with values that are observed in the solar upper chromosphere or that are inferred from observations of other stars. The value for the conductive flux is not so readily observed. A common choice is to set the conductive flux to zero at the loop foot points as well as at the loop summit (e.g. Rosner 1978). Alternatively, one can use the differential emission measure ([FORMULA]) to constrain the values for the conductive flux.

Using this system of equations, one also has to be careful with the initial choice of the value for the loop foot point separation and gradient, X and [FORMULA]. In principle, both the equations describing the arcade and the equations governing the loop allow infinitely long loops. In fact, it appears that a natural choice for [FORMULA] would be to set it to zero. At the loop summit, however, its value tends to infinity. In the case of the arcade this does not pose any real problems as the loop summit is well defined and the loop length is obtained through


If we assume that the loop summit is located at [FORMULA] then the foot point separation, [FORMULA], is given by


As the loop will only deviate very slightly from the arcade at the foot points, we can assume a length for the surrounding arcade and calculate the foot point separation and loop gradient of this arcade. These values are then used as starting points for the loop calculations.

Symmetry imposes a further constraint, namely that the heat gradient along [FORMULA] and hence the conductive flux have to vanish at the loop summit. From this it may seem that the problem is over-determined, but we additionally have to fix the heating. The order of magnitude of the heating can be deduced from the X-ray luminosity of the target stars (see e.g. Collier Cameron 1988).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998