STM-D-1083Paper2007Published and peer-reviewed

The Hall Instability of Weakly Ionized, Radially Stratified, Rotating Disks

Edward Liverts · Michael Mond · Arthur D. Chernin

Abstract and summary · read the original at the source

In one page

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.

Why it matters hereChapter 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.

What it claims

01In 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
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02The 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
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03The 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
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04Two 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
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05The 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)
Published and peer-reviewed
06The 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
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Read it · abstract

Abstract

Cool weakly ionized gaseous rotating disks are considered by many models to be the origin of the evolution of protoplanetary clouds. Instabilities against perturbations in such disks play an important role in the theory of the formation of stars and planets. Thus, a hierarchy of successive fragmentations into smaller and smaller pieces as a part of the Kant-Laplace theory of formation of the planetary system remains valid also for contemporary cosmogony. Traditionally, axisymmetric magnetohydrodynamic (MHD) and, recently, Hall-MHD instabilities have been thoroughly studied as providers of an efficient mechanism for radial transfer of angular momentum and of radial density stratification. In the current work, the Hall instability against nonaxisymmetric perturbations in compressible rotating fluid in external magnetic field is proposed as a viable mechanism for the azimuthal fragmentation of the protoplanetary disk and, thus, perhaps initiates the road to planet formation. The Hall instability is excited due to the combined effect of the radial stratification of the disk and the Hall electric field, and its growth rate is of the order of the rotation period. This family of instabilities is introduced here for the first time in an astrophysical context.

Edward Liverts, Michael Mond and Arthur D. Chernin, The Hall Instability of Weakly Ionized, Radially Stratified, Rotating Disks, The Astrophysical Journal 666, issue 2, 1226–1231 (2007); preprint arXiv astro-ph/0701086.

(Abstract only — see the rights note above. On this site, Rueda, Haisch and Cole on vacuum zero-point field pressure instability in astrophysical plasmas and the formation of cosmic voids is at /library/stm-edaf616bc9, rotating nonlinear magnetic islands in a tokamak plasma — the same question of a rotating magnetised plasma organising itself — are at /library/stm-c728aa7225, and flow generated by electrostatic turbulence in tokamaks is at /library/stm-ea244957db.)

The way in

https://doi.org/10.1086/520489Published as The Astrophysical Journal 666, issue 2, pages 1226 to 1231 (2007), copyright American Astronomical Society, with no open licence recorded, so the sheet is abstract-only and no text beyond the published abstract is reproduced. The authors’ preprint of the same work, arXiv astro-ph/0701086 version 1, 4 January 2007, was fetched and read in full on 2026-09-08; it carries the arXiv assumed-1991-2003 licence rather than a Creative Commons licence, so it does not promote this sheet to an open licence, and every claim below is located to a section, equation or figure of it. The abstract reproduced here is the published journal abstract, which differs from the preprint only in copy-editing and in the correction of two typographical slips. Affiliations at the time: Edward Liverts and Michael Mond, Department of Mechanical Engineering, Ben-Gurion University of the Negev; Arthur D. Chernin, Sternberg Astronomical Institute, Moscow State University. The work was supported by the Israel Science Foundation under contract 265/00.

How 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

Where it sits in the curriculum

Plasmoids, charge clusters and the orbs The unified picture