MHD in weakly-ionised media

Mark WardleMacquarie University

Sydney, Australia

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Outline•Molecular clouds and star formation

•MHD in weakly ionised media

•A comment on the hall effect

•Conductivity in molecular clouds

•Shock waves

•Conductivity in protostellar discs

•Implications for MRI and jets

Introduction

Magnetic fields in molecular clouds •Magnetic fields play an important role during star formation

–Pmag is 30–100 times Pgas in molecular clouds–energy density of magnetic field, fluid motions and self-gravity are similar –field removes angular momentum from cloud cores–field diffusion is required to avoid magnetic flux problem

MHD with finite conductivity

Conductivity

The Hall parameter

•relative strength of magnetic and drag forces in determining drift velocity

particles tied to field

particles tied to neutral fluid

•For ions,

•For electrons,

The conductivity tensor•solving for :

•current density:

– field-parallel conductivity

– Hall conductivity

– Pedersen conductivity

Magnetic diffusion

•If the only charged species are ions and electrons,

•Three distinct diffusion regimes:

– Ohmic (resistive)

– Hall

– Ambipolar

log n

log B

ohmic

hall

ambipolar

Hall effect in fully vs weakly ionised plasma•Hall effect arises through asymmetry in tying of charged particles to

magnetic field lines

•Fully ionised plasma: high frequencies, short wavelengths–difference between ion and electron cyclotron frequencies–ions can no longer keep up with changes to field–short lengthscales often irrelevant at the scales of interest–hall MHD etc

•Partially ionised plasma: low frequencies, long wavelengths–difference between ion and electron collision frequencies–neutral collisions decouple ions before electrons–length/time scale may be comparable to system size/evolutionary time scale–conductivity tensor in generalised Ohm’s law

•How does one reconcile these approaches?–in partially ionised case, ions become attached to the neutrals–effective ion mass is increased by ratio of neutral to ion densities–effective cyclotron frequency is reduced by same factor

•Be careful when estimating relevance of Hall effect!!!

4 6 8 10 12 14-6

-4

-2

0

2log B (G)

log n_H (cm^-3)

electronsions

50 A2500 A

solarnebula

molecularclouds

Umebayashi & Nakano 1990

Nishi, Nakano & Umebayashi 1991

Nishi, Nakano & Umebayashi 1991

3 4 5 6 7 8 9 10 11 12 13 14 15-5

-4

-3

-2

-1

0

1

2

3

Ambipolar

Hall

Ohmic

0.1mlo

g

(1015

cm

km

s-1)

log nH (cm-3)

3 4 5 6 7 8 9 10 11 12 13 14 15-5

-4

-3

-2

-1

0

1

2

3

Ambipolar

Hall

Ohmic

MRNlo

g

(1015

cm

km

s-1)

log nH (cm-3)

3 4 5 6 7 8 9 10 11 12 13 14 15-5

-4

-3

-2

-1

0

1

2

3

Ambipolar

Hall

Ohmiclog

(1015

cm

km

s-1)

log nH (cm-3)

MRNi

3 4 5 6 7 8 9 10 11 12 13 14 15-5

-4

-3

-2

-1

0

1

2

3Ambipolar

Hall

Ohmiclog

(1015

cm

km

s-1)

log nH (cm-3)

MRNx

J-type shock waves

C-type shock waves

Instability

Shock waves - Hall effect

Wardle 1998

Wardle 1998

Chapman & Wardle MNRAS in press

Chapman & Wardle MNRAS in press

Stars, protostellar disks and planets

The minimum-mass solar nebula•Simple model for surface density based on the solar system

(Hayashi 1981)–add H,He etc to each planet to recover standard interstellar abundances–spread matter smoothly

•Resulting surface density:

•Disk mass:

•Useful reference standard

•Temperature is estimated by considering thermal balance for a black solid particle

–assume spherical, radius a, distance r from the sun:

•Then:

– disk is thick beyond 30 AU

– self-gravity negligible

Protostellar disks

Bachiller 1996 ARA&A

Bachiller 1996 ARA&A

Kitamura et al 2002 ApJ

Kitamura et al 2002 ApJ

Kitamura et al 2002 ApJ

Kitamura et al 2002 ApJ

Kitamura et al 2002 ApJ

Kitamura et al 2002 ApJ

Protostellar disks•Role of magnetic field in final stages of formation and subsequent

evolution of protoplanetary discs is unclear–MHD turbulence (magnetorotational instability)?–disc-driven MHD winds?–disc corona?–dynamo activity?

•How strong is the magnetic field?

•Is the field coupled to the material in the disc?–disc is weakly ionised

Protostellar disks are poorly conducting•high density implies low conductivity

–recombinations relatively rapid–drag on charged particles

•deeper layers shielded from ionising radiation for r < 5 AU–x-ray attenuation column ~10 g/cm2 –cosmic ray attenuation column ~100 g/cm2

Igea & Glassgold 1999

cosmic rays

x-ray ionisation rate

How strong is the magnetic field?

•Expect B > 10 mG given the strength in cloud cores

•Compression during formation of disk and star

•Shear in disc may wind up field or drive MRI

•Equipartition field in the minimum solar nebula

•Evidence for 0.1 – 1 G fields in the solar nebula at 1AU

Sano & Stone 2002a

Magnetic field diffusion in protostellar disks•Is the magnetic field coupled to the matter?

< h cs ?

•Which diffusion components dominate?–ohmic, hall or ambipolar?

•What are the consequences of different diffusion regimes?–vector evolution of B shows fundamental differences–hall diffusion reverses sign under global field reversal (yikes)

•Diffusion and magnetocentrifugal jet launching–loading of mass onto field lines–constrains bending of field lines within disk–radial drift of field

•Diffusivity depends on location–vertical stratification of ionisation rate and density–inconvenient radial variations of microphysics

Resistivity calculations

•minimum solar nebula–assume isothermal in z-direction

•ionisation by cosmic rays and x-rays from central star

•simple reaction scheme following Nishi, Nakano & Umebayashi (1993)–H+,H3

+,He+,C+,molecular (M+) and metal ions (M+), e-, and charged grains–extended to allow high grain charge (T larger than in molecular clouds)

•adopt model for grains–none, single size grains, MRN size distribution, MRN+ice mantles,

extended MRN, etc–results for “no grains” or 0.1 m grains presented here

•evaluate resistivity components–when can the field couple to the shear in the disc?–which form of diffusion is dominant?

Ionisation products

Reaction scheme

log

n / n

H (s-1)

M+

C+m+

e

He+

H+H3

+

Abundances: 1AU, no grains

z / h

Criterion for coupling

z / h

log

n / n

H

-14

0

1

2

3-13

(s-1)

-11

-12

M+

-4

-3

-2

C+

m+

e

He+

H+H3

+

Abundances: 1AU, 0.1m grains

5 AU

5 AU

5 AU

5 AU

5 AU

5 AU

Magnetorotational instability (MRI)•magnetic field couples different radii in disc

•tension transfers angular momentum outwards

•kh > 1 required to fit in disc, i.e. vA/cs < 1

•resulting turbulence transports angular momentum outwards

MRI in non-ideal MHD

Ambipolar or ohmic diffusion

Hall diffusion

Wardle 1999

Salmeron & Wardle 2005

Salmeron & Wardle 2005

Salmeron PhD thesis

Stone & Fleming 2003

MRI with dead zone

Sano & Stone 2002b

MRI with hall diffusion

Sano & Stone 2002b

logB2/ 8πP0

Protostellar jets

HH 30

Wardle 1997IAU Coll. 163 (astro-ph)

Wardle 1997IAU Coll. 163 (astro-ph)

Salmeron, Wardle & Königl