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)
Thank you very much for your kind introduction Mr. Chairman.
My name is Hideki Maki from Rikkyo University. And Collaborator is Mr. Susa from Rikkyo Univ.
What I would like to present here is a seed magnetic field strength on accretion disks surrounding first stars considering a dissipation of magnetic flux in primordial gas clouds.
* This work was partly supported by the “Rikkyo University Special Found for Research”.
December
Japan-Italy Seminar Niigata

2. Importance of the First Star’s Mass
Of course, itself is one of the interesting things.
Effect for the reionization of the universe
Massive stars → strong UV → earlier reionization
Heavy elements pollution
Massive stars → production of heavy elements
WMAP
QSO absorption lines
First, I start an importance of the first star’s mass. Of course, itself is one of the interesting things.
The other is the effect for the reionization of the universe. If a first star is very massive, the strong UV is emitted. And then, the reionization epoch is made to be earlier. And this effect is important from the results of WMAP.
It is also important to consider a heavy elements pollution. As you may know that, from the high-redshift QSO absorption lines, we can see that there are the considerable amount of the heavy elements at z~5. Thus, massive stars must be formed in the early universe, then produce the heavy elements.
In brief, a mass of the first stars is important for our to consider the object formation after the first star formed.
The mass of first stars is important to consider
the object formation after the first stars formed.
December
Japan-Italy Seminar Niigata

3. Japan-Italy Seminar 2003 @ 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
perturbation
mass
accretion
stellar core
Next, I will talk how to derive the mass of first stars simply.
This is the 3D simulation of cosmic structure formation. The filament structures appear in this simulation, and there are dense gas regions at the nodes of filaments. And then these dense gas clouds fragment from surrounding gas. After, these fragments contract by self-gravity. So the core and the envelope are formed. Eventually, a final star is formed as the result of the mass accretion to the core.
Therefore, the mass of a star depends on the fragment’s mass and the mass accretion rate.
fragments
(Abel et al. 2001)
A mass of stars depends on the fragment’s mass and the mass accretion rate.
Therefore,
December
Japan-Italy Seminar Niigata

4. Japan-Italy Seminar 2003 @ Niigata
Mass Accretion Rate
accretion disk
stellar core
mass accretion
Envelope with angular momentum
Accretion disk may be formed.
We wish a first star to be massive, then the angular momentum must be transported effectively.
One of the possible mechanism is the turbulence by the magnetorotational instability (MRI) in the present-day star forming region.
(e.g. Sano, Inutsuka & Miyama 1998 etc.)
But, magnetic flux has been neglected on the studies for a first star, because one considers that magnetic flux is very small in the early universe.
Generally, since the envelope seems to have angular momentum, the accretion disk is formed at the mass accretion phase. So, we wish the first stars to be massive, then the angular momentum is transported outside effectively. One of the possible mechanism of the transport of angular momentum is the turbulence by magnetorotational instability (so-called MRI) in the present-day star forming region, which is suggested by the recent studies e.g. Sano, Inutsuka & Miyama 1998.
But, magnetic flux has been neglected on the studies for a first star, because one considers that magnetic flux in the early universe is very small.
Thus, the question rises. What is the mechanism to transport the angular momentum?
Here, I will report that the magnetic flux may not be negligible in the first star formation.
What is the mechanism to transport the angular momentum ?
December
Japan-Italy Seminar Niigata

5. Japan-Italy Seminar 2003 @ Niigata
Purpose
We will study about the angular momentum transport in the primordial case.
As first step, in order to assess the seed magnetic flux in the accretion disk surrounding the first stars, we investigate the appearance of the dissipation of the magnetic flux in the course of collapse of the primordial cloud.
We will study the effect of seed magnetic flux for the transport of angular momentum in primordial cloud.
As first step, in order to assess the seed magnetic flux in the accretion disk surrounding the first stars, we investigate the appearance of dissipation of the magnetic flux in the course of collapse of the primordial cloud.
December
Japan-Italy Seminar Niigata

6. Japan-Italy Seminar 2003 @ Niigata
Method : Contraction
Free-fall core collapse
One-zone approximation
Equation of state
with poritoropic index γ=1.1
We suppose the free-fall core collapse, and the one-zone approximation. Then, we adopt the equation of state with poritoropic index gamma equals one point one, which is approximation for the result of the thermal evolution of the primordial cloud calculated by Dr. Omukai.
(K : constant coefficient.)
December
Japan-Italy Seminar Niigata

7. Method : Chemical Network
Included 23 species
95 chemical reaction network
collision ionization, collision dissociation
recombination, electron attachment
3 body reaction, etc.
no photo-ionization, no photo-dissociation
Next, I’ll talk about chemical network. In order to accurately derive the ionization degree, we include the 23 species relative to hydrogen, deuterium, helium, and lithium. And then we calculate the non-equilibrium 95 chemical reaction network, where the collision ionization, collision dissociation, recombination, electron attachment and three body reaction etc. are included. But, there are no photo-ionization and no photo-dissociation.
Rate equations
December
Japan-Italy Seminar Niigata

8. Dissipation of Magnetic Flux
Dissipation process
Ohmic dissipation
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
Well, here, I will exprain the dissipation of magnetic flux.
There are two dissipation processes. One is the Ohmic dissipation, another is the ambipolar diffusion. First, for the Ohmic dissipation, this phenomena is that magnetic energy is lost as thermal energy by the collision between neutral particles and charged particles.
And next, for the ambipolar diffusion, when magnetic fields exist, the charged particles rotate around the field lines. Here, it is assumed that the direction of gravity is this right arrow. In this case, if charged particles and neutrals is tightly coupled by the collision, then charged particles, neutrals and also magnetic field lines in-fall together. This situation is froze-in. On other hand, if charged particles and neutrals are not strongly coupled, only the neutrals fall into the core, and the charged particles and magnetic field lines are left. In this situation, magnetic field lines seem to come out of the gas. So, this phenomena is called the ambipolar diffusion. B g g B B p e p e p n n e n n
December
Japan-Italy Seminar Niigata

9. Japan-Italy Seminar 2003 @ Niigata
Diffusion Velocity
In this work, the diffusion velocity of field lines from gas is defined using the argument in Nakano & Umebayashi We investigate the degree of dissipation by looking the diffusion velocity compared with the free-fall velocity.
Ohmic dissipation
Ambipolar diffusion
: electrical conductivity
: viscous damping time
notation ν : charged particles
notation n : neutral particles
: reduced mass
: momentum-transfer rate coefficient
Well, I will talk how to estimate the dissipation degree. According to Nakano & Umebayashi, we define the diffusion velocity of magnetic field lines from gas. For the Ohmic loss , its velocity becomes approximately like this, and for the ambipolar diffusion, its velocity becomes approximately like this. And then we investigate the degree of dissipation, comparing this diffusion velocity with the free-fall velocity. In a word, when v_B/u_ff > 1 (v b over u ff is grater than unity), magnetic flux dissipates, and when v_B/u_ff < 1 (v b over u ff is less than unity), magnetic flux is frozen to the gas.
Note that the Ohmic dissipation is independent on the field strength, while the ambipolar diffusion depends on it.
R : radius of cloud, c : speed of light
q : charge, n : number density
: free-fall velocity
M : mass of cloud
December
Japan-Italy Seminar Niigata

10. Results at Present-Day
Evolution of ionized fraction at present-day
diffusion velocity at present-day
evolution of the ionization degree
Here, I’ll present the results of the diffusion velocity in the present-day case.
This figure shows the evolution of fractional abundance for electrons, metal-ions, grains with positive and negative charge, and neutral grains. Horizontal axis denotes the cloud’s density, and vertical axis is the fractional abundance for each particles. The collapse is proceeding to this direction. The evolution of ionization degree is this red curve.
As this result, the calculated diffusion velocity is shown in this figure. Horizontal axis denotes the cloud’s number density, and vertical axis is the magnetic field strength in the cloud. These solid curves show the contours of the diffusion velocity normalized with the free-fall velocity. The diffusion velocity is equal to the free-fall velocity in this line, so you can see that, at lower density than this line, the magnetic flux is frozen to the gas. On the other hand, at higher density than this line, the magnetic flux dissipates from the gas. In this region, the dissipation is defined by the Ohmic dissipation.
frozen
dissipation
(Nakano & Umebayashi 1986)
December
Japan-Italy Seminar Niigata

11. Thermal Evolution of Primordial Cloud
No heavy element and grain
Collapse of the primordial cloud is proceeding at higher temp. than present-day.
The ionization degree may be different from the present-day case.
This figure is the thermal evolution of the gas clouds with various metallicity, which is investigated by Dr. Omukai. Horizontal axis denotes the cloud’s density and vertical axis is the cloud’s temperature. This line (1 Z_sun) shows the thermal evolution in the present-day case, and This line (10^-6 Z_sun) is that in the primordial case. In the present-day case, the collapse is proceeding at ~10 K due to the heavy elements and grains. On the other hand, in the primordial case, the collapse is proceeding at two higher order temperature, because of the absence of the coolant. Thus, the ionization degree in the primordial case may be different from that in the present-day case. So, This dissipation history of the magnetic flux seems to change from the preset-day case!
(Omukai 2000)
Dissipation history of the magnetic
flux seems to be different also.
December
Japan-Italy Seminar Niigata

12. Japan-Italy Seminar 2003 @ Niigata
Initial Conditions
Density :
Temperature :
Mass : Jeans mass :
Chemical abundance :
asymptotic values at low-z
(Galli & Palla 1998)
Initial magnetic flux :
Most uncertain quantity
→ we take the initial magnetic
flux as parameter.　
I will to talk about the initial conditions of calculation.
We set the initial density to equal one thousand per cubic cm, and the initial temperature to equal two hundred fifty Kelvin. For the initial mass, we take the Jeans mass at these initial density and temperature.
For the chemical abundance, we adopt the asymptotic values at low redshift in the results of Galli & Palla 1998.
For initial magnetic flux, is is most uncertain quantityt. So, we take the initial magnetic flux as parameter.
December
Japan-Italy Seminar Niigata

13. Results : Evolution Ionized Fraction
Decreases by
recombination
Ok, I will present the results of our calculations.
This figure is the result of the evolution of fractional abundance for the mainly species, electron, proton, hydrogen atom, hydrogen molecule, lithium, and lithium plus.
Horizontal axis denotes the cloud’s density, and vertical axis is the fractional abundance for each species. The cloud evolves to this direction. At low density, the degree of ionization decreases by the recombination of protons and electrons. But, since the recombination rate of the Lithium plus is smaller then that of the protons, the decrease of the ionization degree is almost stopped. And then, at high density, the ionization degree increases because of the high temperature.
It is understood from this that the decrease of the ionization degree is limited at asymptotic ten to the power of minus twelve.
Next, as this result, I will present the diffusion velocity.
December
Japan-Italy Seminar Niigata

14. Results : Diffusion Velocity
Dissipation
frozen
Similarly, horizontal axis denotes the cloud’s density, while vertical axis is the magnetic field strength.
This red curves show the contours of the logarithmic value of the diffusion velocity normalized with the free-fall velocity. This line is minus four, and this line is just zero. Thus, at this region upper this line, magnetic flux dissipates from the gas. And, at this region under this line, magnetic flux is frozen to the gas.
This blue line shows the magnetic field strength Bcr (b critical) which satisfies the condition that gravitational energy is comparable to magnetic energy. Therefore, since we are interesting to a collapsing cloud, the magnetic flux must be smaller than this line. Furthermore, as you can see that this Bcr line is always lower this line, which is that the diffusion velocity equals the free-fall velocity. In other words, the magnetic fields are always frozen to the primordial gas, as long as we consider the collapsing cloud.
For example, the magnetic flux which is 10^-11 G at the initial density grows along this green line in the frozen-in condition. In this case, at expected densities of the accretion disk formed, the flux is amplified to >~10^-5 G ( greater than or asymptotic ten to the power of minus five Gauss).
December
Japan-Italy Seminar Niigata

15. Japan-Italy Seminar 2003 @ Niigata
MRI
Condition in which the MRI can grow
Growth timescale by MRI > Dissipation timescale
(Tan & Blackman 2003, astro-ph/ )
If initial flux
If initial flux
When the growth timescale is greater than the dissipation timescale, the MRI can grow. According to Tan & Blackman 2003, the minimum field strength are derived as this equation. Its minimum strength is asymptotic 10^-5 G at this density. And, this density is here. So, to drive the MRI, the initial field strength must be greater than 10^-11 G. Therefore, if the initial flux is grater than 10^-11 G, the MRI can be driven. On the other hand, if initial flux is smaller than 10^-11 G, it can not be driven. Furthermore, we can expect that the angular momentum is transported in this configuration.
→ MRI can be driven
→ MRI can not be driven
December
Japan-Italy Seminar Niigata

16. Japan-Italy Seminar 2003 @ Niigata
Conclusions
Magnetic fields are frozen, as far as the condition that B < Bcr ~ 10-5(nH/103 cm-3) G is satisfied.
If Bini >10-11 G at nH=103 cm-3, magnetic flux might be amplified to B G at nH cm-3.
* In this case, the MRI seems to be driven also in the primordial case.
To conclude, I would like to summarize my talk.
First, magnetic fields are frozen, as far as the condition that flux is smaller than B critical is satisfied.
Second, If capital b is equal to ten to the power of minus eight Gauss at the density is equal to ten to the power of three per cubic cm, magnetic flux might be amplified to be greater than or asymptotic to ten to the power of minus two Gauss at the density is greater than or asymptotic to ten to the power of thirteen per cubic cm.
Finally, as I said before, the initial magnetic field strength is uncertain quantity. But, the value from Langer et al is largest in the reported values by many authors. So, this estimated value for seed magnetic flux on the accretion disk is the upper limit. According to recent studies in the present-day case e.g. Sano, Inutsuka, & Miyama 1998 or Tan & Blackman 2003, if the magnetic flux is >~ 10^-4 G, the MRI could be driven, then angular momentum seems to be transported also in the primordial case.
December
Japan-Italy Seminar Niigata