## 2. Loop model## 2.1. The coordinate systemSince 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 where . The vector perpendicular to the field is given by
## 2.2. The equation of motionThe equation of motion for a plasma where fluid motions are much slower than the sound speed is given by: where where is the height (above the stellar surface) divided by the stellar radius. Along the field lines, i.e. in the direction of , vanishes so that the equation of motion reduces to where is the component of g along the field lines. In the isothermal case this equation can be integrated to yield ([JC]) where is given by Here, and are the surface ratios of the gravitational and centrifugal energies to the thermal energies, i.e. where is the Boltzmann constant and 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 Along , the direction perpendicular to the field lines, the equation of motion becomes This essentially describes the balance of forces due to magnetic tension (RHS), pressure and gravity. ## 2.3. Energy balanceThe energy loss function is given by where is the heat flux due to particle
conduction, is the ratio of the specific heats,
and denotes the radiative heat losses. We
explore two different expressions for the sources of heating that have
been collected together in . Firstly,
, where is the scale
height in stellar radii and 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 where is the conductive flux and is given by For a fully ionized plasma, is of the order
of W m In an adiabatic situation, the total change in heat is zero so that . We can hence write © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |