Rigidly Rotating Magnetosphere model
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The Rigidly Rotating Magnetosphere (RRM) model builds on the foundations established in the Magnatically Confined Wind Shock (MCWS) model, by considering the ultimate fate of the post-shock plasma near the tops of closed magnetic loops. As discussed by Babel & Montmerle (1997), in the absence of rotation the plasma will soon fall back to the star under the downward pull of gravity. These downflows are seen quite clearly in the MHD simulations of θ1 Ori C (a star that does rotate, but not at a sufficiently rapid rate for the centrifugal force to play any role).
However, in the presence of significant rotation, any post-shock plasma situated outside the Kepler co-rotation radius (defined for aligned dipole systems as the circle in the common equatorial plane on which gravitational and centrifugal forces cancel) will be supported against the inward pull of gravity by the outward centrifugal force. As long as the magnetic confinement remains significant, the plasma can sit in stable equilibrium - and with steady wind feeding, accumulate - over extended periods of time, leading to the establishment of a magnetosphere that rotates rigidly with the star (hence the name of the model!). The details of the RRM model are presented in Townsend & Owocki (2005).
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Key aspects
The fundamental assumption of the RRM model is that eveywhere the magnetic field may be treated as rigid - that is, the model corresponds to magnetic confinement in the idealized limit 
. This assumption means that the model cannot realistically be applied to moderately-confined stars such as θ1 Ori C (
). However, for stars such as σ Ori E (
), the model works very well out to significant distances ( > 10R * ) from the star. In fact, the RRM model outperforms MHD simulations in this latter scenario, since MHD in the limit of large η becomes impractical due to the very large Alfvén velocity.
The advantage of the rigid-field assumption [also adopted in the MCWS and Rigid Field Hydrodynamics (RF-HD) models] is that determining the evolution of the post-shock plasma becomes, in essence, an analytical problem - one that can be solved using very modest computational resources. The key aspect is that the plasma is acted on by many forces (radiative, gravitational, magnetic, inertial), but only two of these exhibit any significant component in the direction along each magnetic field lines, and therefore only two - specifically, the gravitational and centrifugal forces - can influence the plasma's movement. (In fact, the radiative force can in principle exert an influence; but because the post-shock plasma is slow-moving, the Sobolev optical depth is typically too large for radiative driving to be efficient).
Further simplifications come from the fact that the combined gravitational and centrifugal forces can be expressed as the gradient of the scalar effective potential
Then, finding out where post-shock plasma will tend to end up, for each individual field line, reduces to the problem of finding local minima in Φeff (as sampled along the field line; these local minima represent points of stable equilibrium). Together, the minimum-potential points for all field lines define a locus known as an accumulation surface.
Accumulation surfaces
Animations of dipole-field accumulation surfaces, for differing values of the magnetic obliquity β, are presented below. The surfaces are truncated at some finite outer radius, in order to better show their geometry.
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Observe how, with increasing β, the accumulation surfaces become deformed away from a planar disk morphology. Accompanying the deformation is the appearance of detached 'leaves', seen quite clearly in the β=80° animation. These leaves eventually join the main surface when β=90°, to form a pair of truncated cones centered on the rotation axis.
Plasma distribution
Although accumulation surfaces show the points of local equilibrium throughout a star's circumstellar environment, not all of the post-shock plasma will end up on these surfaces. This is simply a consequence of the finite temperature and hence pressure pressure of the plasma; although the plasma cools significantly after undergoing the shocks, the radiation field of the star ensures that it never gets too cold. Assuming a fixed temperature T * for all of the plasma, the plasma distribution in the vicinity of an accumulation surface may be calculated simply by integrating the equation of hydrostatic equilibrium,
Here, ρ(s) is the density at some arc distance s along a field line from the equilibrium position s = 0 (the latter being defined by Φeff' = 0,Φeff'' > 0). To fix the normalizing density ρ(0) requires some model for the wind upflow that is filling up the star's magnetosphere (see Townsend & Owocki 2005 for details).
Animations of the plasma distribution associated with the foregoing accumulation surfaces are presented below. The color scheme in these animations alternates between one based on the Doppler velocity of the plasma, and one that shows (in false color) the column density of the plasma.
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With the steady feeding of plasma from the wind, magnetospheric densities will grow linearly with time, following the distributions shown in the animations above. Eventually, however, so much material will accumulate that the rigid-field assumption finally breaks down. As this point is approached, field lines will begin to sag outward, and eventually break open, spilling the magnetospheric plasma via a centrifugal breakout episode.
Application to σ Ori E
The RRM model has proven very succesful in reproducing the optical variations exhibited by σ Ori E. The provisional model for the star (see Townsend, Owocki & Groote 2005 for full details) adopts a magnetic obliquity β=55° and an inclinaiton i=75°, and assumes a rotation angular frequency Ω * of 50% of the critical value. It is further assumed that the magnetic origin is displaced by 0.3R * from the center of the star, in the direction perpendicular to both axes; this offset enables the model to reproduce the asymmetries seen in the observations. Results from this model, and its comparison with the observations, are presented in the images and animations below.
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