3D MHD simulation ofprimordial star forming cloud

Kenji Eric Sadanari(Tohoku Univ.)

Kazu Omukai(Tohoku Univ.)
Kazu Sugimura(Meryland Univ.)
Tomoaki Matsumoto(Hosei Univ.)

Introduction

Magnetic ﬁeld in star-formation
Star formation process

Jet or outﬂow

Gas cloudBProtostar
Pseudo disk

Collapse phase

Magnetic eﬀect
・magnetic braking
・magnetic dissipation
・outﬂow/jet

Protostar formation

Accretion phase・Formation of circumstellar disk & binary
・Star formation eﬃciency
・mass of star

Magnetic eﬀect is important for star formation processMagnetic ﬁelds in the Primordial gas
Seed magnetic ﬁeld
・Observable limitation by CMB→B < 1 × 10−9G・Theoretical model(Biermann battery)→ B ∼ 10−30 − 10−20G→ this B-feld is too small to have inﬂuence on star formation

Ampliﬁcation of B-ﬁeld
by turbulence(small scale dynamo)

Strong magnetic ﬁelds
there is a possibility to change
ﬁrst star formation process

B-ﬁeld is important in primordial gas too

Schober et al.2012

Problem of previous MHD simulation
Machida et al.2008

Rotation

EOS: barotropic relation P ∝ ρ γ obtained
from one-zone model w/o B-ﬁeld
In reality, T-evolution can deviate
from barotropic relation byshock heating andmagnetic eﬀects:
slow down of collapse by B-ﬁeld
magnetic dissipation heating

Energy eq & chemistry should be solved
consistently in MHD simulation.
B-ﬁeldTemperature evolution with magnetic eﬀect
Nakauchi et al .2019(one-zone model)
Eﬀect of Lorentz forcetﬀtcol =1 − (B/Bcr)2B ⤴ ➡ tcol ⤴
➡deviation of T-evolution ⤴
Parameter of cloud structureB ∝ ραsphere : α = 2/3 sheet-like : α = 1/2

T-evolution depends on collapse time(tcol) & cloud shape(α)
3D MHD simulation is neededMethodMHDシミュレーション
Set-upof 3D MHD simulation
SFUMATO (Matsumoto.2007)
・self-gravitational MHD code
・AMR
・Energy eq( + cooling/heating )[Basic equation]@⇢+ r · (⇢~v ) = 0@t

r2 = 4⇡G⇢<latexit sha1_base64="4Rmdf/5v2Bc/evW48ICnpjm81AQ=">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</latexit><latexit sha1_base64="hktFax/9aBfIBSx2itGJTQE7SqY=">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</latexit>@~v⇢+ ⇢(~v · r) =@t<latexit sha1_base64="vG9674vJiJ+Q76S2t1epSRcHIZM=">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</latexit>rP

1 ~~B ⇥ (r ⇥ B)
4⇡

⇢r∂e1
+∇⋅e + p + | B |2 v − B (v ⋅ B) = Λ)∂t2[(]<latexit sha1_base64="MI29QJ0W0DMFlYH7TrWCak/mu58=">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</latexit>~@B~= r ⇥ (~v ⇥ B)@t
Ideal-MHD

Λ: cooling from H2, HD lines, chemical, H2, H- continuum[chemical reaction: 27 reactions among 9 species]
H, H2, e, H+, H−, H+2, D, HD, D+

MHDシミュレーション
Set-upof 3D MHD simulation[Set-up]・critical Bonnor-Ebert sphere

1.4

・initial center density n0 = 103[cm−3]
・using initial Temperature & chemical abundance of one-zone
・no rotation・uniform magnetic ﬁeld

γ=Emag| Egrav |= 0 ∼ 9 × 10−1Result−2Eﬀect of magnetic ﬁelds : case of γ = 4 × 10
TModerate B-ﬁeld
γ=Emag| Egrav |= 4 . 0 × 10−2

No rotationnB

−2Eﬀect of magnetic ﬁelds : case of γ = 4 × 10
T

Shockn−2Eﬀect of magnetic ﬁelds : case of γ = 4 × 10
T

ShocknThermal evolution at center

T[K]n[cm−3]T-evolution changes only slightly even in strong B-ﬁeld

Magnetic evolution at center
B∝ρ
B

α=1

B
Bcrα
①α = 2/3

Shock②
α = 1/2

α = 2/3③α=0α=1α = 2/3

α = 1/2
α=0④
α=1
Shock

n[cm−3]

Gas collapse progresses due to the inﬂow along B-ﬁeld

Summary & Future work
I investigated the eﬀect of the B-ﬁeld on the thermal evolution ofprimordial gas using 3D MHD simulation which solves energy eq& chemistry
Gas collapse progresses due to the inﬂow along the B-ﬁeld
and thermal evolution did not change much from one-zone modelThis MHD simulation was able to reproduce shock heating
formed on the surface of the pseudo disk
[future work]
Starting from a rotating magnetized cloud and examine the eﬀectof angular momentum transport and jet/outﬂow
Study on eﬀect of Magnetic dissipation heating by non-ideal MHD
simulation .