## 3. Details of the modelWe 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 . 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 . This allows us to change
the scale-height of the external field simply by varying 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
. We ignore this effect when calculating the
loop equilibrium, but calculate 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 yields (18). Eq. (20) is a rewrite of Eq. (10). ## 3.1. The surrounding plasmaTo 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 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 and , as the shape of the arcade is fixed by our choice of a potential two-dimensional field for , Note that because of the assumption of a potential field, the shape
prescription will not be valid for large values of the ## 3.2. The loop equilibriumThe 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 is small compared to unity we just recover the potential field solution. So the loop shape will only differ from the arcade if 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.
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 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, becomes negative and starts to approach values of the order of unity. Eventually, the RHS of Eq. (28) becomes negative and will have to increase. ## 3.3. The boundary valuesTo 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 () 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, If we assume that the loop summit is located at then the foot point separation, , 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 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). © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |