Historical research guide · 1083 · 2007
The Hall Instability of Weakly Ionized, Radially Stratified, Rotating Disks
Edward Liverts · Michael Mond · Arthur D. Chernin
In one minute
The disk of gas and dust around a young star is cold, barely ionised and turning. Liverts, Mond and Chernin ask what happens in such a disk when you keep the one term most models throw away: the Hall electric field, which appears when the electrons slip free of the ions and the neutral gas and drift on their own. They find a new instability. It does not need the shear that drives the famous magnetorotational instability; it needs only the disk’s radial layering plus the Hall drift, and it grows in about one orbit. It also does not stay axisymmetric — it breaks a smooth ring of gas into lumps around the circle, each roughly as big across as the disk is thick. The authors put numbers to it for a real protoplanetary disk and argue those lumps are planetesimals, the first solid bodies. Their closing point is the neat one: it is precisely because the gas is so poorly ionised that the effect turns on at all.
Connection
Chapter 9 is about plasma that organises itself into structures and holds them, and here the organiser is the Hall drift in a gas so faintly ionised that most models treat it as neutral — a reminder that a very small charged fraction can still run the dynamics. Chapter 13 gets a clean astrophysical case where the plasma term everyone drops turns out to set the outcome, in this instance the first step toward building planets.
From the source
In a cool, weakly ionised, radially stratified rotating disk threaded by an axial magnetic field, the Hall electric field acting together with the radial stratification excites non-axisymmetric instabilities whose growth rate is of the order of the rotation period. The authors state that this family of instabilities is introduced here for the first time in an astrophysical context.Abstract; Section 2.3
The Hall term matters only when the length scale of the inhomogeneity is shorter than the ion inertial length divided by the square root of the ionisation degree. In a protoplanetary disk the only ionisation sources are non-thermal — cosmic rays, X-rays and radioactive decay — giving an ionisation degree near ten to the minus twelfth, so that length runs up to about one astronomical unit and the Hall model, not ideal magnetohydrodynamics, is the correct one for realistic radial density structure.Section 2.1, Eq. 4; Section 3, disk parameter estimates
The instability does not depend on rotational shear, unlike the magnetorotational instability. In the local dispersion relation it enters through a single parameter that measures the Hall drift against the stratification length, and as that parameter goes to zero the ordinary magnetohydrodynamic result — stable fast magnetosonic waves combined with epicyclic oscillations — is recovered.Section 2.2, remark after Eq. 5; Section 2.3, Eq. 14 and the limit for small alpha
Two distinct unstable regimes are identified. In the first, density and total pressure fall radially in the same direction with the pressure falling faster, which happens automatically in a polytropic disk; numerical solution for Keplerian rotation gives instability for the ratio of sound speed to Alfven speed between one and twenty. In the second, the two gradients have opposite signs — as in a young disk with radial inflow, or in rings left by gravitational instability — and rotation is stabilising, but the growth rate keeps rising as the magnetic field grows stronger.Section 2.3, cases 1 and 2, Eqs. 16 to 18 with Figs. 1 and 2
The unstable branch is a quasi-electrostatic slow mode: the perturbations in ion density and ion velocity do the work, while the perturbed electron density and the perturbed magnetic field are negligible. That is what makes it a genuinely Hall effect rather than a variant of a magnetohydrodynamic mode.Section 2.3, closing paragraph, citing Liverts and Mond, Physics of Plasmas 11, 55 (2004)
The proposed consequence is that radial density rings in a protoplanetary disk break azimuthally into fragments of size comparable to the disk thickness within about one rotation period, and that these fragments are the planetesimals from which planets are later assembled. The bounding condition on the wavenumber, combining survival against thermal pressure with the validity of the linear analysis, is satisfied for typical disk values — again because the ionisation degree is so small.Section 3, Eq. 19 and the closing discussion
The way in
https://doi.org/10.1086/520489How to cite it
Edward Liverts, Michael Mond, Arthur D. Chernin (2007) The Hall Instability of Weakly Ionized, Radially Stratified, Rotating Disks doi:10.1086/520489