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

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2. Loop model

2.1. The coordinate system

Since we are principally interested in the vertical extent of closed loops, we model a region close to the equator, where the centrifugal force has the greatest effect on the loop height. For loops that emerge at higher latitudes, the centrifugal force no longer acts in the plane of the loop but tends to distort it, pulling it towards the equator. The effect of this distortion will be the subject of another paper. Here, we follow the setup and coordinate system chosen by [JC]. The loop lies along the x -direction and the y -axis points radially outwards (see Fig. 1). If the loop summit is at [FORMULA] and the foot points are at [FORMULA], the vector along the magnetic field can be described by

[EQUATION]

where [FORMULA]. The vector perpendicular to the field is given by

[EQUATION]

[FIGURE] Fig. 1. The coordinate system and definitions for the loop.

2.2. The equation of motion

The equation of motion for a plasma where fluid motions are much slower than the sound speed is given by:

[EQUATION]

where p and [FORMULA] are the gas pressure and density. Here [FORMULA] combines the effect of gravity and the centrifugal force and is given by

[EQUATION]

where [FORMULA] is the height (above the stellar surface) divided by the stellar radius. Along the field lines, i.e. in the direction of [FORMULA], [FORMULA] vanishes so that the equation of motion reduces to

[EQUATION]

where [FORMULA] is the component of g along the field lines. In the isothermal case this equation can be integrated to yield ([JC])

[EQUATION]

where [FORMULA] is given by

[EQUATION]

Here, [FORMULA] and [FORMULA] are the surface ratios of the gravitational and centrifugal energies to the thermal energies, i.e.

[EQUATION]

where [FORMULA] is the Boltzmann constant and [FORMULA] is the hydrogen mass.

Integrating the equation of motion over a pill-box that encompasses the boundary field line, one finds that the gas and magnetic pressure are conserved across the field lines, hence

[EQUATION]

Along [FORMULA], the direction perpendicular to the field lines, the equation of motion becomes

[EQUATION]

This essentially describes the balance of forces due to magnetic tension (RHS), pressure and gravity.

2.3. Energy balance

The energy loss function is given by

[EQUATION]

where [FORMULA] is the heat flux due to particle conduction, [FORMULA] is the ratio of the specific heats, and [FORMULA] denotes the radiative heat losses. We explore two different expressions for the sources of heating that have been collected together in [FORMULA]. Firstly, [FORMULA], where [FORMULA] is the scale height in stellar radii and A is the loop cross-section. This allows us to vary the overall magnitude of the heating and its scale height. Secondly, we have investigated heating functions that are proportional to the gas density.

For strong magnetic fields conduction works mainly along the field lines. We take the form developed by Spitzer (1962) for the conductive heat flux with

[EQUATION]

where [FORMULA] is the conductive flux and is given by

[EQUATION]

For a fully ionized plasma, [FORMULA] is of the order of [FORMULA]  W m-1  K [FORMULA]. The radiative heat loss scales like [FORMULA], where [FORMULA] is the electron density. For a fully ionized plasma this it is related to the temperature and pressure by [FORMULA]. [FORMULA] is taken from Rosner et al. (1978) who obtained a piecewise power-law fit to the loss function calculated by Raymont et al. (1976). For higher temperatures we use the extension given in Collier Cameron (1988).

In an adiabatic situation, the total change in heat is zero so that [FORMULA]. We can hence write

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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