Hideki Maki

1
Dissipation of Magnetic Flux in Primordial Clouds

• Hideki Maki
• Department of Physics, Rikkyo University

Collaborator Hajime Susa (Rikkyo University)
This work was partly supported by the Rikkyo
University Special Found for Research.
2
Importance of the First Stars Mass

• Of course, its own is one of the interesting
  things.
• Influence on the reionization of the universe
• Massive stars ? strong UV ? earlier reionization
• Heavy elements pollution
• Massive stars ? production of heavy elements

WMAP
QSO absorption lines
The mass of first stars affects the formation of
objects after the first star forming.
3
Mass of First Stars

• Star formation
• Fragments of high dense gas that are the parent
  of star are formed.
• Mass accretes to the core that is formed at the
  center of the fragment.

envelope
stellar core
mass accretion
CDM density perturebation
fragments
filament
4
Mass Accretion Rate
Envelope with angular momentum
Formation of accretion disk
Accretion rate depends on the rate of angular
momentum transport.

• Mass of the first star depends on the mechanism
  of angular momentum transport.
• One of the possible mechanism is the turbulence
  by magnetorotational instability.

(e.g. Sano, Inutsuka Miyama 1998 Sano
Inutsuka 2001)
But, we are neglecting the magnetic fields on the
studies for first star, since one consider that
magnetic flux in the early universe is very weak.
Magnetic flux must be considered in primordial
cloud too?!
5
Thermal Evolution of Primordial Cloud
Collapse of primordial gas cloud
No heavy elements and grains
Collapse at high temperature
Much differential thermal history than present-day
(Omukai 2000)
Is dissipation history of the magnetic flux
different ?
6
Purpose

• In order to assess the seed magnetic flux in the
  accretion disk of first stars, we investigate the
  appearance of dissipation of the magnetic flux in
  the course of collapse of the primordial cloud.
• We will study the influence of seed magnetic flux
  for the transport of angular momentum in
  primordial clouds.
• We will derive the mass of first stars.

7
Method Contraction

• Free-fall core collapse
• One-zone approximation
• Equation of state
• with poritoropic index ?1.1

(K constant coefficient.)
8
Method Chemical Network
Non-equilibrium of 24 species
114 chemical networks

• collision ionization, collision dissociation
• recombination, electron attachment
• 3 body reaction, etc.
• no photo-ionization, no photo-dissociation

Rate equations
9
Dissipation of Magnetic Flux
Dissipation process

• Ohmic loss
• Magnetic energy loss as thermal energy, by
  charged particles colliding with neutral
  particles.
• Ambipolar diffusion
• Charged particles twine around the magnetic lines
  by Lorenz force. On the other hand, since
  neutrals are not influenced by the Lorenz force,
  neutral particles fall into the stellar core with
  the relative velocity for charged particles. So,
  magnetic flux seems to diffuse out the cloud at
  the point of neutrals.

n
n
e
n
p
e
p
e
e
p
n
p
n
n
B
B
B
p
e
p
e
p
n
n
e
n
n
10
Diffusion Velocity
In this work, the relative velocity of
dissipation for collapse gases is defined using
the argument in Nakano Umebayashi 1986. We
investigate the degree of dissipation by looking
a ratio of diffusion velocity for free-fall
velocity.

• Ohmic loss
• Ambipolar diffusion

electrical conductivity
viscous damping time
notation ? charged particles
notation n neutral particles
reduced mass
momentum-transfer rate coefficient
R radius of cloud, c speed of light
q charge, n number density
free-fall velocity
M mass of cloud
11
Results at Present-Day
Evolution of ionized fraction at present-day
diffusion velocity at present-day
main charged particles e, ions
dissipation
frozen
main charged particles charged grains
(Nakano Umebayashi 1986)
12
Initial Conditions of Primordial Cloud

• Density
• Temperature
• Mass Jeans mass
• Chemical abundance
• values at 1z 1 (Galli Palla 1998)

13
Initial Magnetic Flux
rotation of protogalaxy
Still too much uncertainty.
generation of vorticity
generation of seed magnetic flux (Biermann
battery effect)
We regard the initial magnetic flux as parameter.
We take B 10-25-105 G.
dynamo by rotation of galaxy within galaxy
formation time
(Pudritz Silk 1989)
14
Results Evolution Ionized Fraction
15
Results Diffusion Velocity
Dissipation
Viscous dumping time
Frozen
Cyclotron frequency
Resign of ambipolar diffusion
Resign of ohmic loss
Br2const.
16
Inefficient Ohmic Loss
diffusion velocity at present-day
diffusion velocity at primordial
Ohmic loss efficient
main charged particles grains
(Nakano Umebayashi 1986)
17
Conclusions

1. Magnetic fields are frozen, as far as the
   condition Blt10-5(nH/1 cm-3) G is satisfied.
2. If B10-20 G at nH1 cm-3, magnetic flux might be
   amplified to B10-6 G at nH1018 cm-3.

future
We next study the influence of seed magnetic flux
for the transport of angular momentum in
primordial clouds.