4. Numerical results
We used a Runge-Kutta method (Press et al. 1986) to solve the system of dependent differential equations. For a given set of initial values the equations were integrated until the loop flattened off; we required that the change in height was less than . We have employed two different approaches to find solutions to Eqs. (15- 21, 23- 25); in the first approach, we prescribe the length of the arcade and then adjust the heating until the conductive flux vanishes at the loop summit (we required to conductive flux to fall below 10-7 of the initial flux). Alternatively, we can specify the heating and then adjust the loop length until the conductive flux vanishes at the loop top. For some heating values there are two loop lengths that will have vanishing conductive flux at the summit. In this case, the solutions with the longer loop lengths will show a temperature inversion, i.e. the maximum temperature is reached below the loop summit.
We also calculated the value of along the loop and rejected solutions with much greater than unity.
4.2. Loop solutions
We find that there are two distinct families of boundary conditions that yield solutions with vanishing conductive flux at the loop summit. It is possible to start with either a high or a very low initial value for the conductive flux. This simply corresponds to setting the base of the loop at slightly different levels in the transition region. Starting with a high initial conductive flux is very similar to following the behaviour of the low-flux solution, once the flux maximum has been reached. For loops without temperature inversions, it is in fact easier to prescribe the conductive flux (as this is an observable quantity through the DEM) at the loop foot points and then to adjust the heating until a solution is reached where vanishes at the loop summit.
We have investigated different parameterisations for the heating function. In the first approach we chose an exponential heating with , where A could either vary as an inverse function of the magnetic field strength or could be kept fixed. In this case the loop and arcade summit temperatures will differ if either the value of differs in- and outside the loop, or, alternatively, if the conductive flux differs. If the heat ratio between the loop and the arcade are changed, one usually also has to adjust the ratio of the base pressures so that the conductive flux reaches zero at the loop and arcade summit. Note that even if A is kept constant along the loop, the shape variation is still included in the equation for the conductive flux (19) through the term as the area of the loop varies proportionally to .
In the second approach we parameterised the heating as . One slightly worrying aspect of this parameterisation is that the heating will start to increase outwards once the loop rises above the co-rotation radius and the gas density starts to increase. For either heating parameterisation we find that the loop shape depends mainly on the ratio between the loop and arcade pressure. In the following sections we describe the behaviour of over- and under-pressured loops.
4.3. Over-pressured hot loops
Loops that are hotter and have higher pressure than the surrounding arcade can be produced by increasing the conductive flux inside the loop. We generally find that the foot points of high-pressure loops come closer together when the loop starts to deviate from the arcade shape. This is because the magnetic tension needs to increase in order to contain the loop as , the ratio between the gas and magnetic pressure, increases. This increase in the magnetic tension is achieved by decreasing the loop foot point separation. Thus, in Figs. 3 and 4 the curves on the left-hand side of the diagram are for the over-pressured loops.
Under-dense loops: These can arise with either heating parameterisation, but only when one deals with large temperature gradients. For low temperature gradients, the difference in loop and arcade summit temperature is not large enough to offset the pressure difference and one tends to get loops with an under-dense region close to the foot points where the temperature gradient is very large and a denser loop summit.
For high temperature gradients (i.e. large initial conductive flux values), the temperature difference at the loop summit becomes so large that the gas in the loops is less dense than in the arcade along the whole length of the loop. Fig. 2 shows the shape, temperature, pressure and density for a typical hot and over-pressured loop with high conductive flux at the base. It was obtained by increasing the base conductive flux inside the loop by a factor of 1.5 with respect to the base flux in the arcade.
If there is no pressure difference between the loop and arcade foot points and if the star rotates rapidly enough with loops that rise above the co-rotation radius, there is usually a height above which the external pressure becomes larger than the internal pressure. This happens in particular if the temperature in the loop is much higher than in the arcade, so that the pressure rise in the loop is dampened much more than the pressure rise in the arcade. At the pressure cross-over, the pressure difference in (see Eq. (29)) can for a while become less important than the density difference, so that there is a range of loop lengths and field strengths where over-pressured loops have larger foot point separations than the arcade.
The dotted lines in Fig. 3 show the loop summit height as a function of foot point separation for high-pressure and under-dense loops on a star with solar and 50 times solar rotation rate. The solid lines are for loops that have the same initial conductive flux, pressure and temperature as the arcade.
Over-dense loops: If we start off with a higher base pressure inside the loop, the resulting loop will usually be denser than the arcade everywhere. For an over-dense loop, the pressure and density terms in Eq. (29) have opposite signs below the co-rotation radius, but the density term will never be strong enough to offset the effects of the pressure difference. The foot points of hotter, and more importantly, over-pressured loops will therefore be closer together than for the arcade.
The dotted line in Fig. 4 shows the way an over-dense and over-pressured loop will be deformed as for a star with solar and 50 times solar rotation. The solid lines show the summit height for loops with the same base pressure and heating as the arcade. Whereas the heating was proportional to the gas density in Fig. 3, we used an exponential heating law for Fig. 4. Comparing the thicker lines (i.e. solar rotation) in the two figures shows that the heating parameterisation only has very little effect. The differences between the loop heights of the faster rotator are not due to the heating but to the fact that the over-pressured loops in Fig. 4 are also over-dense whereas the over-pressured loops in Fig. 3 are under-dense (see Sect. 4.6).
4.4. Over-pressured cool loops
A cool high-pressure loop is formed when the loop pressure is enhanced and the conductive flux is suppressed. Due to their higher pressures, the foot point separations of these loops usually decrease as they rise. In some cases, however, the higher conductive flux in the arcade and the resulting temperature increase can lead to a slower pressure fall-off in the arcade until the external pressure exceeds the internal pressure. This means that the loop foot points can be overpressured and the loop summit underpressured compared to the surrounding plasma. In these cases the foot point separations will increase as the loops rise.
4.5. Under-pressured and hot loops
Under-pressured loops tend to show increasing foot point separation as they rise and as the relative importance of the magnetic field decreases. Under-pressured and hot loops are just the opposite of the over-pressured cool loops described in 4.4. They are produced when the base pressure in the loop is suppressed with respect to the arcade pressure and when the conductive flux is enhanced. Again, we can get a pressure cross-over when the higher loop temperature slows the pressure fall-off in the loop sufficiently so that the loop pressure can fall below the arcade pressure.
For over-pressured loops, however, the values for the summit height as a function of loop foot point separation should only be taken as rough guides. This is because we assume a potential magnetic field for the surrounding arcade. But the loop will only start to deviate from the arcade shape once becomes of the order of unity inside the loop. For an under-pressured loop will generally be larger on the outside, so that the assumption of a potential field for the surrounding plasma tends to break down before the loop starts to deviate from the arcade shape.
One interesting distinction between the over-pressured and under-pressured loops are their maximum summit heights. Over-pressured loops have a natural maximum loop height as , which is given by , has to remain positive. For an over-pressured loop the internal field strength is always less than the external field strength and will finally reach zero when . Note that this can only happen once exceeds unity. Under-pressured loops in general have no such natural summit height, but for under-dense loops there is an additional constraint that follows from the requirement that has to remain negative. This means that has to hold (see Eq. (18)). This is a weaker constraint as is usually much smaller than .
4.6. Under-pressured and cool loops
Plasma in hot under-pressured loops is always less dense than the surrounding plasma, but whether the plasma in cool and under-pressured loops is over- or under-dense depends on the ratio of the foot point pressures and on the temperature gradient.
Over-dense loops: Most loops with high but suppressed conductive flux values will be over-dense independent of the heating parameterisation. The loop summit height as a function of the foot point separation is shown in Fig. 3 (dashed line) for heating that is proportional to the gas density. Below the co-rotation radius, the pressure and density terms act in the same direction (see Eq. (29)). Above it, the density term changes sign and tries to bring the foot points closer together. As before, it is usually the pressure difference that decides the fate of the loop and in most cases, the loop foot point separation will increase. Very fast rotators are an exception to this, as there is a narrow range of loop lengths and magnetic field strengths where the buoyancy term is larger than the pressure difference and for which the loop foot points move together.
Under-dense loops: Cool and under-pressured loops where the gas at the summit is less dense than in the arcade tend to arise when the gas pressure at the base is higher in the arcade than in the loop. They can also occur when the gas pressure is the same at the loop and arcade base but the conductive flux in the loop is suppressed with respect to the arcade. In this case the resulting loops will always be cool and under-pressured, but only loops with comparatively low values of the base conductive flux will be under-dense.
For under-dense and under-pressured loops, the pressure and density difference act to pull the loops in opposite directions below the co-rotation radius, though the pressure term is usually orders of magnitude larger. Above the co-rotation radius, both terms act to pull the loop foot points apart. This is in contrast to what we observe in over-dense and under-pressured loops where the pressure and the density term counteract above the co-rotation radius.
The importance of Eq. (29) and the difference between the over-dense and under-dense loops can be seen clearly when comparing the loop heights for the fast rotator in Figs. 3 and 4. Whereas the loop heights for the solar rotation rate are similar on both figures, the maximum loop heights for the rapid rotator are of the order of 2.7 in Fig. 3 and of the order of 2 in Fig. 4. The loops in Fig. 3 can rise higher as the pressure and density terms have opposite signs once the loops reach above the co-rotation radius.
4.7. The conductive flux and heating at the base
There is a threshold value for the conductive flux above which the loop summit temperature is almost uniquely defined by the conductive flux at the base. This threshold value is a function of the base pressure.
Below the threshold value, it is mainly the heating that determines the loop summit temperature. Maximum summit temperature is reached for very low heat input and, perhaps contrary to intuition, the loop summit temperature starts to fall as the heating is increased and we may get temperature inversions along the loop. Some examples are shown in Fig. 5.
Not all solutions with a temperature inversion are physical as the conductive flux can not always reach zero for a second time. This may be because of the expansion of the loop that is included in our model. The energetics for a variety of different loops, including some `inverse-gravity' loops, have been discussed in detail in van den Oord & Zuccarello (1996). In agreement with these authors, we find that the minimum amount of heating required in order to obtain a solution where the conductive flux vanishes at the summit increases as the rotation rate increases, and that lower heating produces hotter loops.
4.8. The magnetic field strength and scale height
For a given magnetic field configuration and strength, the loops attain a maximum summit height. Fig. 6 shows the loop summit height as a function of the foot point separation for over-pressured loops and four different values of . At first the loop follows the arcade shape very closely, but as the ratio of the gas to the magnetic pressure increases, the magnetic tension has to increase so that the gas can be contained at the loop summit. The distance between the loop foot points therefore starts to decrease, though the loop summit height does not increase significantly. The turn-off point also depends strongly on the difference between the gas pressure inside and outside the loop. The larger the difference in gas pressures, the earlier the loop shape will deviate from the arcade shape.
We can also vary the value of k that determines the fall-off of the magnetic field. As k increases, the magnetic field strength falls off more rapidly and the loop summit height decreases. If shear is introduced (, where ), the external field falls off less rapidly with height. The effect is therefore similar to lowering k without changing the foot point separation.
The effects of varying the scale height on the maximum loop height are shown in Fig. 7.
As it is the ratio between the magnetic and the gas pressure that determines the loop height and the turn-off point, raising the gas pressure at the base has a similar effect as lowering the magnetic field strength in so far as the loop shape is concerned. The energetics of the loop, however, will be rather different. For fast rotators, the relative importance of the density term in Eq. (29) increases. Depending on whether the pressure and density terms have opposite signs above the co-rotation radius, the loop height either increases or decreases. Fig. 8 shows the effect of increasing the stellar rotation rate on the loop summit height for an over-pressured and under-dense loop. As the density term tries to counteract the effects of the pressure term above the co-rotation radius, the loop turnoff is delayed increasingly for higher rotation rates. The figure also shows that the influence of rotation is not very strong, so that we need to take into account two further effects: it is thought that the strength of the magnetic field increases with rotation rate, an effect that would allow the loops to rise higher. But if total pressure balance is maintained, the increase of the magnetic field might also go hand in hand with an increase in the base pressure, an effect that will shorten the loops.
For fast rotators, most length-dependent scaling relations (see eg. Rosner et al. 1978) break down. This is because the pressure scale height, and hence the pressure structure of the atmosphere, depends critically on the rotation velocity.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998