Rigid Field Hydrodynamics model

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The RF-HD model for σ Ori E

The Rigid Field Hydrodynamics (RF-HD) model uses a similar approach to the Rigidly Rotating Magnetosphere (RRM) model, to predict the magnetospheric plasma distribution of stars whose fields remain essentially rigid. However, unlike the simple hydrostatic formalism applied in the RRM model, the RF-HD model uses time-dependent hydrodynamics to simulate plasma upflow, shocks and accumulation along field lines. The resulting simulations allow predictions to be made of a star's X-ray and UV properties - much like full 3-D MHD simulations, but at a fraction of the computational cost.

1 Key aspects
- 1.1 Geometry
- 1.2 Body Forces
2 Implementation
3 Application to σ Ori E
4 See also

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Key aspects

As with the RRM model, the magnetic field lines are assumed to remain completely rigid. Then, each field line may be considered as a 1-dimensional conduit for plasma flow, with a cross-sectional area that varies inversely with the magnetic flux density. Since there is no cross-communcation between these conduits, the problem of modeling the flow along a group of N field lines reduces to N independent 1-D hydrodynamical simulations of flow in the presence of fbody forces.

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Geometry

For a dipole magnetic field, the space curve described by a given field line is defined by the parametric equations

$r(\theta) = r_{0} \sin^{2}\theta, \qquad \phi = \mbox{const.};$

here, $r 0$ parameterizes the radius of the field line summit. The natural spatial co-ordinate to use for describing the flow along the field line is the arc length $s$ , defined by the differential equation

${\rm d}s = r_{0} \sin\theta \sqrt{1 + 3\cos^{2}\theta}\,{\rm d}\theta.$

Integrating, this becomes

$s(\theta) =\frac{r_{0}}{2} \left[-\frac{\sinh^{-1}(\sqrt{3}\cos\theta)}{\sqrt{3}} - \cos\theta\sqrt{1 + 3\cos^{2}\theta}\right],$

where the convention is that $s = 0$ at the field-line summit $θ = π / 2$ . The magnetic flux density along the field line is given by

$B(\theta) = \frac{\sqrt{1 + 3\cos^{2}\theta}}{2 \sin^{6}\theta}\,B_{0},$

where, likewise, $B 0$ is the summit flux density. Hence, by conservation of flux, the area element must vary as

${\rm d}A(\theta) = \frac{2\sin^{6}\theta}{\sqrt{1 + 3\cos^{2}\theta}}\,{\rm d}r_{0} \,{\rm d}\phi,$

and the volume element is given by

${\rm d}V(\theta) = {\rm d}A(\theta)\,{\rm d}s = 2 \sin^{7}\theta \,r_{0} {\rm d}r_{0} \,{\rm d}\theta \,{\rm d}\phi.$

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Body Forces

There are three forces that act on the plasma as it flows along a field line: gravitational, centrifugal and radiative (the Coriolis force is always directed perpendicular to the flux tube, and therefore is balanced by the magnetic tension). Assuming for simplicity that the dipole field considered previously is aligned with the stellar rotation axis, then the accelerations due to the gravitational and centrifugal forces are given by

$\mathbf{g}_g = - \psi \frac{G M_{*}}{r^2}$

and

$\mathbf{g}_{\Omega} = \frac{3}{2} \psi\, \Omega^2 r \sin^{2}\theta,$

respectively, where

$\psi \equiv \hat{\mathbf{r}} \cdot \hat{\mathbf{s}} = \frac{2\cos\theta}{\sqrt{1 + 3\cos^{2}\theta}}$

is the projection of the unit radial vector $\hat{\mathbf{r}}$ onto the unit vector $\hat{\mathbf{s}}$ locally tangent to the field line.

Likewise, the radiative force within the CAK formalism is given, for a point-source star, by

$\mathbf{g}_{\rm cak} = \frac{F_{\rm cak}}{r^{2}} \left(\frac{\delta_{v}}{\rho}\right)^{\alpha}.$

The important quantity in this expression is

$\delta_{v} = \hat{\mathbf{r}} \cdot \nabla (\hat{\mathbf{r}} \cdot \mathbf{v}) = \hat{\mathbf{r}} \cdot \nabla (\psi v),$

the radial gradient of the projected flow velocity $ψ v$ in the radial direction (see, e.g., Owocki 2004 for a definition of the other symbols).

To calculate $δ v$ , we first note that $ψ$ depends only on $θ$ , and therefore commutes with the directional derivative $\hat{\mathbf{r}} \cdot \nabla$ . Hence, we have

$\delta_{v} = \psi \hat{\mathbf{r}} \cdot \nabla v.$

Then, we express the gradient operator in terms of derivatives along and across the field line,

$\nabla = \hat{\mathbf{s}} \frac{\partial}{\partial s} + \hat{\mathbf{t}} \frac{\partial}{\partial t}.$

(we have ignored any azimuthal component to the gradient, since it cannot contribute to $δ v$ ). Here, $\hat{\mathbf{t}}$ is the unit vector in the cross-field direction,

$\hat{\mathbf{t}} = \frac{\hat{\mathbf{s}} \mathbf{\times} (\hat{\mathbf{r}} \mathbf{\times} \hat{\mathbf{s}})} {|\hat{\mathbf{r}} \mathbf{\times} \hat{\mathbf{s}}|},$

and $t$ is the corresponding cross-field coordinate. Thus, we obtain

$\delta_{v} = \psi^{2} \frac{\partial v}{\partial s} + \sqrt{1 - \psi^{2}} \frac{\partial v}{\partial t}.$

Since the flow along each field line is simulated independently of the others, we lack the information to calculate the cross-field gradient embodied by the second term in the above expression. Hence, we make the ansatz that this term is zero, giving

$\delta_{v} = \psi^{2} \frac{\partial v}{\partial s}.$

Note that in cases where $δ v$ would be negative (e.g., in the presence of downflows), it should instead set to a sensible value such as zero.

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Implementation

To simulate the 1-D hydrodynamical flow along each field line, Rich uses the VH-1 hydro code, a finite-volume code written by John Blondin and co-workers that is based on the Piecewise Parabolic Method introduced by Colella & Woodward (1984). Standard RF-HD models (if there is such a thing) include 250 points along each field line, distributed non-uniformly so that the region below the sonic point, at the field-line footpoints, is adequately sampled. Body forces are included using the formalism given above, and radiative cooling is modeled using the cooling function from MacDonald & Bailey (1981).

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Application to σ Ori E

For obvious reasons, Rich chose σ Ori E as the first star to test the RF-HD model on. So far, three models have been applied to the star - each having the same fundamental parameters (e.g., mass, radius, rotation rate), but with differing time- and space-sampling strategies:

Base run: 2,550 field lines (50 $φ$ values from 0 to $2π$ , 51 $r 0$ values from $1.2\,R_{\rm p}$ to $7.45\,R_{\rm p}$ ), 200 rotation periods starting from an 'empty' magnetosphere, 5,000 time points stored.
4-cycle run: 4 rotation periods continuing from the end of the base run, 1,000 time points stored.
Extension run: repeat of the base run, but with an additional 25 values in the $r 0$ direction, going out to $10.575\,R_{\rm p}$ , to give a total of 3,800 field lines.

The 4-cycle run is designed to study the rotational modulation of a 'full' magnetosphere, while the extension run will furnish the basis for future 4-cycle runs that include the outer parts of the magnetosphere (these parts are responsible for the hard X-ray emission). Animations of the results from the base and 4-cycle runs are presented below.