1. Hideki Maki Department of Physics, Rikkyo University
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”.
September
Mini Niigata

2. Importance of the First Star’s 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.
September
Mini Niigata

3. Mini workshop @ Niigata
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
CDM density
perturebation
mass
accretion
stellar core
fragments
mass of a star depends on the fragment’s mass and mass accretion rate.
Therefore,
September
Mini Niigata
filament

4. Mini workshop @ Niigata
Mass Accretion Rate
accretion disk
stellar core
mass accretion
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?!
September
Mini Niigata

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 ?
September
Mini Niigata

6. Mini workshop @ Niigata
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.
September
Mini Niigata

7. Mini workshop @ Niigata
Method : Contraction
Free-fall core collapse
One-zone approximation
Equation of state
with poritoropic index γ=1.1
(K : constant coefficient.)
September
Mini Niigata

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
September
Mini Niigata

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 n p e e p e e p n p n n B B B p e p e p n n e n n
September
Mini Niigata

10. Mini workshop @ Niigata
Diffusion Velocity
In this work, the relative velocity of dissipation for collapse gases is defined using the argument in Nakano & Umebayashi 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 ν : 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
September
Mini Niigata

11. Results at Present-Day
Evolution of ionized fraction at present-day
diffusion velocity at present-day
main charged
particles : e, ions
frozen
dissipation
main charged
particles :
charged grains
(Nakano & Umebayashi 1986)
September
Mini Niigata

12. Initial Conditions of Primordial Cloud
Density :
Temperature :
Mass : Jeans mass :
Chemical abundance :
　 values at 1+z =1 (Galli & Palla 1998)
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Mini Niigata

13. Mini workshop @ Niigata
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 = G.
dynamo by rotation of galaxy
within galaxy formation time
(Pudritz & Silk 1989)
September
Mini Niigata

14. Results : Evolution Ionized Fraction
Decreases by
recombination
September
Mini Niigata

15. Results : Diffusion Velocity
Dissipation
: Viscous
dumping time
Frozen
　：Cyclotron
frequency
Resign of
ambipolar
diffusion
Resign of
ohmic loss
Br2=const.
September
Mini Niigata

16. Inefficient Ohmic Loss
diffusion velocity at present-day
diffusion velocity at primordial
Ohmic loss efficient
main charged
particles : grains
(Nakano & Umebayashi 1986)
September
Mini Niigata

17. Mini workshop @ Niigata
Conclusions
Magnetic fields are frozen, as far as the condition B<10-5(nH/1 cm-3) G is satisfied.
If B=10-20 G at nH=1 cm-3, magnetic flux might be amplified to B～10-6 G at nH=1018 cm-3.
future
We next study the influence of seed magnetic flux for the transport of angular momentum in primordial clouds.
September
Mini Niigata