分子雲コアの角運動量・磁場構造と原始惑星系円盤の形成
Angular momentum and magnetic ﬁeld
structure of cloud coresand formation of protoplanetary disks

Sanemichi Takahashi(Kogakuin University/NAOJ）

Formation and Evolution of Protoplanetary Disks
Gravitational collapse of cloud core

Formation of protostar
and protoplanetary disks

Dust grows in the disk.
Planet formation

Angular momentum problem
Assuming cloud core mass = Msun, r=0.1pc,
angular velocity 0.3 km s-1 pc-1
centrifugal radius ~400 au
Gas cannot collapse to the central star directory
Gas make the disk and accrete onto the star
due to the angular momentum redistributionin (or out of) the disk.
infalling gas
Angular momentum of coreis important star, disk, andplanet formation.

Outline•

Angular momentum of cloud cores•

Collapse of cloud cores and disk formation••

without magnetic ﬁeld•

with magnetic ﬁeld

Analytic model of collapse of the cloud coreOutline•

Angular momentum of cloud cores•

Collapse of cloud cores and disk formation••

without magnetic ﬁeld•

with magnetic ﬁeld

Analytic model of collapse of the cloud coreRotation Velocity ProﬁleCaselli et al. 2002：Velocity gradient(ex. rigid rotation: v=rΩ, dv/dr=Ω)ES
INRigid
DARK
CLOUDS.
XIV.
rotation
DARK
CLOUDS.
XIV. likeComplex proﬁle
255

255

15, 2016

Angular Velocity Distribution

Figure 3. The disc lifetime tlife as a function of !0 (upper), Menv (middle)
and LX,0 (lower). The dotted lines show the dependence of disc lifetime onthese parameters.

26 cores (Caselli et al. 2002, cf Goodman et al.1993)(Kimura Kunitomo Takahashi 2016)

Figure 4. The distribution function for !0 . The thin line shows the obser-1 pc-1]
Angularvelocity:
0.1-6
[km
s
vational data and the thick line shows the fitting result.

Typically 1 [km s-1 pc-1] (?)

Angular Momentum of Cloud Coressize-velocity relation

36Angular momentum transport during the formationand early evolution of stars - EES2012(Belloche2013,

cf Goodman et al. 1993, Ohashi et al. 1997)

Outline•

Angular momentum of cloud cores•

Collapse of cloud cores and disk formation••

without magnetic ﬁeld•

with magnetic ﬁeld

Analytic model of collapse of the cloud core

Observation of Infalling Envelope
rotation velocity proﬁlesL1527 5
-1

Rotational Velocity (km s )Disk and envelope structures of L1527 IRS

pin=0.50

1Figure 8. PV diagrams of the C18O (J = 2–1) emission in TMC-1A along (a) the major axis and (b) the min Figure 1, PA = 73°). These PV diagrams have the same angular and velocity resolutions as those ospacing from 3σ (1σ = 7.1 mJy beam1). Central vertical dashed lines show the systemic velocity (Vsycentral position. Points with error bars in panel (a)9are mean positions derived along the position (vertic
velocities along the outﬂow-axis direction for i = 70° and 40°.

TMC1-Ato be kinematically quite similar to those around T
Tauri stars (Guilloteau & Dutrey 1998; Simon et al.
2000; Rodriguez et al. 2010). A tantalizing questionis whether this disk is also geometrically similar to
those around T Tauri stars. Because this disk is almost
edge-on, it is also possible to investigate the verticalstructures of the disk. In this subsection geometricalstructures of the Keplerian disk are investigated.
4.2.1. Continuum Visibility distribution

pout=1.22

50

100

To investigate the disk structures, we have performed
direct model fitting to the observed continuum visibility
data, which is free from any non-linear eﬀects associatedwith interferometric imaging. Figure 6 shows distributions of the continuum visibilities in three panels, where
red and blue points denote the same groups; the red
Figure 9. Mean positions of the◦ PV diagram along the major axis plotted on apoints are located near the
major axis (±15 on the uvlog R−log V plane. The ordinate is not deprojected. Blue and red points show
blueshifted
redshiftednear
components,
respectively. Dashed lines show theplane) while the blue points
are andlocatedthe minor
best-ﬁt lines with a double power law. In addition to the inner and outer power,
axis. Note that all the thevisibilities“break” radius ofand eachvelocity baseline
are included asin
free parameters. The best-ﬁt
parameters are Rb = 67 AU, Vb = 2.4 km s1, pin = 0.54 ± 0.14,
each track for ∼ 30 minutes
averaged in these plots.
andpareout = 0.85 ± 0.04. The error for (Rb, Vb) is ∼1%.
That is why any trajectory due to Earth’s rotation is not
6a).theItbreakshoulddrawn on the uv-plane (Figure
By using
pointbe(Rbstressed, Vb) = (67 AU, 2.4 km s1), thedynamicalof
massof thecentral protostar
that although all the visibilities
eachbaseline
are av-and the speciﬁc angularmomentum at Rb, which are not inclination corrected, can be
eraged, no further azimuthal
average has been done incalculated at M*p  M* sin2 i  0.43 M: and jp  j sin i Thisis because
our analysis as is obvious7.7inqFigure
104 km6a.
s1 pc,respectively.
We should note, however,
the areanalysiscertain representative
points on a PV
information on structuresthat
thatnotusing
spherically
symdiagram
presented
here
may
systematicallyunderestimateRb
metric such as disks is missed with azimuthally averaged
when the spatial resolution is not high enough, as discussed invisibilities unless the disks
are face-on,
as explained
be-consideration, the real
Appendix
A. Takingthis point into
low in more detail.
Figure 6b exhibits a trend that the visibility am8
plitudes are higher at shorter uv-distances. The total flux density 176 mJy measured in the image space(Figure 1) appears consistent with an amplitude atzero uv-distance, which can be derived from visual extrapolation of the amplitude distribution. Figure 6b(Aso et al. 2017,
et a. 2014)
Figure 5. Mean positions of the PV diagram cf
alongOhashi
theRadius (AU)vrotmajor axis plotted on a log R − log V plane. The ordinate is
not deprojected. Blue and red points show blueshifted andredshifted mean positions in Figure 4a, respectively. Dashedlines show the best-fit lines with a double power law.

-pr(Aso et al. 2015)

Innerregion:p~0.5→
Kepleriandiks
dius of the break and the inner power-law index. The
radius of the identified Keplerian disk can be estimatedOuter: p~1 (j~const)from the best-fit break radius.
Note thatregion
low angularresolutions along the minor axis of disks cause emission
within the radius to be measured at a given velocity→ Infalling envelope
channel and the emission moves the representative position inward at the channel, as pointed out by Aso et al.
(2015). This underestimation strongly aﬀects in the caseof edge-on configurations, such as L1527 IRS, becausesuch configurations make the angular resolution evenlower along the minor axis. We thus take into accountthe underestimation. In their notation, the correction

Dynamics of the envelope is directory observed.

break radius cbreak radiusand jp ∼ 1 ×

In the prediagram cutti
characterize thof a Keplerian
section, phyinvestigatedmaps. To qu
TMC-1A, wechannel maps
Dutrey et al.
with these obthat the obse
diskbut also
motions as drequired that
dominant sho
shown in Figu
law index clo
emission in t
arise from theﬁt the channe
disk models.
We use a d(2014). The msummarized
temperaturedescribed as

4 (R ) Constant j in Infalling Envelope
Takahashi et al. 2016
Is this region formed by the infalling gasthat conserves j ? (cf. Li et al. 2014)
Two ﬂuid elements that conserve j

j1 < j2

infall toward
central star

j1 < j2

Even when the infalling gas conserves j, the constantspeciﬁc angular momentum region does not appear.

re,i ∼ 2000 [AU]

Numerical Simulation andAnalyticModel−18ρc ∼ 9.1 × 10
g cm−3

-2
-2

10
10

10-3

10-3
10

-3

10-4 -4

10

-1

10

1

Specific Angular Momentum [pc km s-1]-1

Velocity [km s ]Specific Angular Momentum [pc km s ]-1 kSpecific
AngularMomentums-1]
Specific
AngularMomentum[pc km s[pc]Collapse of the cloud core (withoutmagneticṀ = 2.2
× 10−6 M⊙ﬁeld)
yr−1
3D
numerical
simulation(Machidaetal.
−6 2010)
−1
-2
Model
Ṁ
=
6.6
×
10
M
yr
⊙
10
Simulation
Analytic model (Takahashi et al. 2013, j2016)
ini
vφ =

Model
SimulationSimulation

10-2 Model10-4
101

j =angularjini
Speciﬁc
−0.5
v
∝
r
φ
momentumis almost
−1
v
∝
r
φ
constant
in

10-3

initial j proﬁle

100AU ! r ! 1000AU

Model
100
1000
10000
Simulation x-direction
RadiusSimulation[AU] z-direction

10

r

10-4

4

Constant10j1 region is formed Model
Simulation x-direction
Model
0
10
Simulation
z-directionevenwith the initiallyrigid-rotating
core
Simulation x-direction
Simulation z-direction

Origin of “constant j region”
Analytic model：
Conservation of j in the envelope is assumed
j~constrini~const
Star formation：Run-away collapse
prolongate the envelope radially.

j1

j ~ const

j2

Run-awaycollapse

j1 ~ j2

The region with constant speciﬁc angular momentum
can be formed as a consequence of strong
prolongation in a run-away collapse.

Observation of Infalling Envelope
rotation velocity proﬁlesL1527 5Figure 8. PV diagrams of the C18O (J = 2–1) emission in TMC-1A along (a) the major axis
in Figure 1, PA = 73°). These PV diagrams have the same angular and velocity resolution
spacing from 3σ (1σ = 7.1 mJy beam1). Central vertical dashed lines show the systemic v
central position. Points with error bars in panel (a) are mean positions derived along the pos
velocities along the outﬂow-axis direction for i = 70° and 40°.

9-1

Rotational Velocity (km s )Disk and envelope structures of L1527 IRS1

pin=0.50

TMC1-Ato be kinematically quite similar to those around T
Tauri stars (Guilloteau & Dutrey 1998; Simon et al.
2000; Rodriguez et al. 2010). A tantalizing questionis whether this disk is also geometrically similar to
those around T Tauri stars. Because this disk is almost
edge-on, it is also possible to investigate the verticalstructures of the disk. In this subsection geometricalstructures of the Keplerian disk are investigated.
4.2.1. Continuum Visibility distribution

pout=1.22

50

100Radius (AU)Figure 5. Mean positions of the PV diagram along the
major axis plotted on a log R − log V plane. The ordinate is
not deprojected. Blue and red points show blueshifted andredshifted mean positions in Figure 4a, respectively. Dashedlines show the best-fit lines with a double power law.

To investigate the disk structures, we have performed
direct model fitting to the observed continuum visibility
data, which is free from any non-linear eﬀects associatedwith interferometric imaging. Figure 6 shows distributions of the continuum visibilities in three panels, where
red and blue points denote the same groups; the red
Figure
9. Meanpositionsdiagrampoints are located near the
major
axis
(±15of◦ theonPVtheuv- along the major axis plotted on alog R−log V plane. The ordinate is not deprojected. Blue and red points show
plane) while the blue points
are locatednearcomponents,
the minor
blueshifted
and redshifted
respectively. Dashed lines show the
best-ﬁt
lines
with
a
double
power
law.
In
additionaxis. Note that all the visibilities of each baseline in to the inner and outer power,
the “break” radius and velocity are included as free parameters. The best-ﬁteach track for ∼ 30 minutes
are averaged
67 these
AU, Vbplots.
= 2.4 km s1, pin = 0.54 ± 0.14,
parameters
are Rb = in
= 0.85± 0.04.rotation
The error forout tob, Vb) is ∼1%.
That is why any trajectoryandpdueEarth’sis(Rnot
drawn on the uv-plane (Figure 6a). It should be stressed1
By using
the breakpoint (Rthat although all the visibilitiesof eachbaselineareavb, Vb) = (67 AU, 2.4 km s ), thedynamical
masshasof thecentralprotostareraged, no further azimuthal
averagebeen
done
in and the speciﬁc angularmomentum at Rb, which are not inclination corrected, can bebecauseour analysis as is obviouscalculated
in Figureat6a.
2 i  0.43 M and
M*pThis M*issinjp  j sin i :information on structures7.7
that
are
not
spherically
sym
4

1
q 10 km s pc, respectively. We should note, however,
metric such as disks is missed
withazimuthally
averagedthat theanalysisusing certain
representative points on a PV
presented
may systematicallyunderestimate Rb
visibilities unless the disksdiagram
are face-on,
as hereexplained
bewhen the spatial resolution is not high enough, as discussed in
low in more detail.
Appendix A. Taking this point into consideration, the realthat the visibility amFigure 6b exhibits a trend
plitudes are higher at shorter uv-distances. The to8
tal flux density 176 mJy measured in the image space(Figure 1) appears consistent with an amplitude atzero uv-distance, which can be derived from visual extrapolation of the amplitude distribution. Figure 6b(Aso et al. 2017,
cf Ohashi et a. 2014)

vrot(Aso et al. 2015)

r-p p=0.85<1: j of infalling gas decreases

dius of the break and the inner power-law index. The
radius of the identified Keplerian disk can be estimatedfrom the best-fit break radius. Note that low angularresolutions along the minor axis of disks cause emission
within the radius to be measured at a given velocity
channel and the emission moves the representative position inward at the channel, as pointed out by Aso et al.
(2015). This underestimation strongly aﬀects in the caseof edge-on configurations, such as L1527 IRS, becausesuch configurations make the angular resolution evenlower along the minor axis. We thus take into accountthe underestimation. In their notation, the correction

Magnetic braking ?

p=1.22>1: j of infalling gas increases ???

brea
brea
and

In
diag
char
of a
secti
inve
map
TMC
chan
Dutr
with
that
disk
moti
requ
dom
show
law
emis
arise
ﬁt th
disk
W
(201
sum
temp
desc

where M is the mass of the molecular clou
2.1: Relation between the velocity dispersion
and thesize of 2.2
the molecular
cloud.
Figureshows cloud.
the observed mgure is taken from Heyer & Brunt (2004).
1/2
µ
≡
2πG
M/Φcloud . The mass
intrinsic
Mass-to-Flux ratio
1
critical mass to flux ratio.

Magnetic Fields in Cloud Core

>1 supercriticallog

intrinsic

0.5

0<1 subcritical

-0.5

-1

(Heiles & Crutcher 2005)
-1.5

21

21.5

22

22.5

23

23.5

24

log N(H2)

Mass-to-ﬂux
ratio
is widely
distributed
2.2: The
normalized mass toflux ratio.
The horizontal
axis is the column density
molecular clouds. µintrinsic > 1 is the super critical and µintrinsic < 1 is the subcritical.
around critical value.
gure is taken from Heiles & Crutcher (2005).

Collapse with Magnetic Field
Angular momentum transfer in collapse phase(Magnetic Braking)
Magnetic"ﬁeld
Cloud core296

"
Lorentz force+
Rotation

Magnetic ﬁeld transports the angular momentum
upward.The angular momentum of the infallinggas decreases

Magnetic Braking Prevents Disk Formation?
Centrifugal balance: j2/r3 = GM/r2

r=j2/GM

Disk forms when j>0.
If magnetic braking eﬃcient and j<(GMr)1/2 is satisﬁed,
disk is not formed.
The Astrophysical Journal, 738:180560(19pp), 2011 September 10

M. N. Machida
al.
Li,et
Krasnopolsky,& Shang[Vol.
63, 810:L26 (5pp), 2015 September 10
The Astrophysical Journal
Letters,

Tsukamoto et al.

The eﬀect of magnetic braking on the disk formationis still under debating.
turbulence, misalignment, non ideal MHD
1022

Ambipolar

1020

Magnetic diffusivities (cm2s-1)

Magnetic diffusivities (cm2s-1)

1022

Hall

1018
1016
1014
1012

Ambipolar

1020

Hall

1018
1016
1014
1012

Ohmic

1010

104

106

108
nH(cm-3)

1010

1012

104

106

108
nH(cm-3)

1010

1012

(cf. Seifried et al. 2012, Joos et al. 2012, Matsumoto et al. 2004)

Figure 3. Absolute values of the computed magnetic diffusivities for the MRN (left panel) and large grain (right panel) cases, for the illustrative magnetic field–density
relation given in Equation (5). For the MRN case, the Hall diffusivity ηH is negative everywhere.

-19.0

-18.0

-17.0

-16.0

-15.0

-14.0

pointmass= 1.1375878e+33
frame=600-13.0

1•1016

-19.0

-18.0

-17.0

-16.0

-15.0

pointmass= 1.1375878e+33
frame=600

-14.0

-13.0

2•1016
-18

•10

1.0

5•1015

1•1016-170

Downloaded from http://pasj.oxfordjournals.org/ at Library of Research Reactor Institute, Kyoto University on A

Ohmic

1010

1.0•100-17

1.0•10Figure 4. Magnetic diffusion coefﬁcients, IO, I H, and IA at the center as afunction of the central density in Model Para. The red line shows IO , the green
line shows I H where the dashed line shows the region of I H  0 , and the solidline shows the region of I H  0 , and the blue dash–dotted line shows IA . The
black thin line shows the “critical value” of I H , 3 q 10 20 Bc (cm2 s1)
suggested by Krasnopolsky et al. (2011) above which the disk is formed intheir simulations. Here, Bc is the central magnetic ﬁeld.

Figure 2. Solid lines show the ratio of the sum of the centrifugal force and the-5•1015

pressure gradient force to the radial gravitational force, q1 

-1•1016
1.0•

to the radial gravitational force, q2 
0-2•1016
5.0•1015

1.0•1016

1.5•1016

2.0•1016-2•1016.

-1•10160

1•10162•1016vG2 x
r '. The dash–dotted lines showq  0.5. In the regions where the dashed lines are larger than the dash–dottedlines, the gas is mainly supported by the centrifugal force. The top and bottom
panels show the results of Models Para and Ortho, respectively. The epochs ofeach model are the same as those in Figure 1.

Figure 4. Density distribution (color map) and velocity field (white arrows) of the reference model (Model REF) in the meridian (left panel) and equatorial (rightpanel) planes, at a representative time t = 6 × 1012 s. The highly flattened, dense equatorial structure is not a rotationally supported disk, but rather a magneticallysupported, nearly non-rotating pseudodisk. Also plotted in the left panel are poloidal field lines, with the same magnetic flux between adjacent lines. The color bars
Fig. 3. Density distribution (color and contours) and velocity vectors (arrows) on the equatorial plane at four different epochs for model 1. The elapsed
above the panels are for log(ρ), with g cm−3 and cm as the units for ρ and length.

(Li et al. 2011)

r p Sr 'Here, vG , p, and Φ are the rotation velocity, gas pressure, and the gravitational
potential, respectively. The dashed lines show the ratio of the centrifugal force

10 -18-1•1016vG2 x(Machida et al. 2011)

time t , protostellar mass Mps , circumstellar disk mass Mdisk , and envelope mass Menv are listed in each panel. The dashed circle in each figure
corresponds to the density contour for n = 1011 cm!3 above which the Ohmic dissipation begins to become effective.

Figure 5. Cross-section of vG in the x–z plane in Model Para. The epoch of thesnapshot is the same as that in Figure 1.

(Tsukamoto et al. 2015)
outside of the parental core. Thus, the angular momentum of
the direction opposite to the disk would eventually be castOutline•

Angular momentum of cloud cores•

Collapse of cloud cores and disk formation••

without magnetic ﬁeld•

with magnetic ﬁeld

Analytic model of collapse of the cloud coreAnalytic Model of Infalling Envelope◉Investigate the eﬀect of the magnetic ﬁeld
from another pint of view of numerical simulations
We already develop the model
without magnetic ﬁeld (Takahashi et al. 2013)◉Advantage of the analytic modelCalculation of long term evolution(Kimura, Kunitomo, Takahashi 2016)
Parameter survey
Comparison with observations (Takahashi et al. 2016)…

The analytic model is useful to investigate
formation and evolution of disks磁場入り星形成モデル

al.Collapse(2016) の式 (5)より、ofMolecular
Cloud
Core
の高さ
! (Takahashi" et al."2013,#
2016)
Assumption：1r
1! dr = − 2GMln+−
Spherical
collapse,
Isothermal,z = 2ridt
∆ri
f ri
ri
r
(cf. Cassen and Moosman 1981)

Pressure gradient force
れを使うと

u

r-1

。各時刻 t での ∆r = (dr/dri )∆ri , ∆φ = (dφ/dri )∆ri (磁力線で

(
!
"
#
$
%
We calculatethemassinfall
rate
−1
&
'
&
'
r
i
れ) が分かれば磁力線の形状が分かる。
etral.
(2016
Ωi
ri 2
ri
1 Takahashiritaking into account= approximately
ln
+
−
1
2
r
2GM
f
ri
r
sineﬀectθi rof pressure.
the
"%1 1
−2
2Ω2i tﬀ
x dx
dx= t = 2tﬀ!)
−11− 1π sinπ θi x x f −1
ln ln
x+
f −1
x x+−1x−定でrについて差分をとると、
iri : initial radius θ = π/2 より
こで x = r/r . 赤道面を考えると

x

1

i

dx

i

tff : free2 fall time

1% 1

∂x−2
x ∆rdx!
− tﬀ ! 2Ωi tﬀ
i) − 1 ∂ri
π φgravity/pressrue= f −1 ln x + x−1
f −1 ln x + x−1 −f :1Initialπxf −1 ln x + x−1

Eﬀect of Magnetic Fields
We use ideal MHD in the envelope and aligned ﬁelds
Magnetic ﬁelds deforms with collapse
magnetic tension
magnetic ﬁelds

magnetic tension

Magnetic Tension
Neglecting the back reaction（~upper limit of magnetic tension）

We focus on midplane gas
We obtain time evolution ofangular momentum andderive condition fordisk formation approximately.

magnetic tension

!

"Condition for Disk formation
2
B0i

ji
ab −a/2−1
ji − j(x) = −
ln x
− e
> −e
(83
8π GM
r ρi /ri2
磁場入り星形成モデル
以下2
!
"2
$
%
）. (Ωi ri )GMr /ri$ GM! r /r2i"%−1
2 2
exp −8π(84
1!> 16π e " 2
2 B0i
GMr /ri
B0i /ρia=ab −a/2 B0i /ρi
GMr −
ρi /r
W − e
= −a/2 + ln(ab/2)8π− ln(a/2ln(ab/2))i
2 ln(x)) （W が多価関数、x < 0 で W < −1 となる方の解x) = ln(−x) − ln(−とする。角運動量が 0 になる半径は、、解析モデルから得られる円盤半径はA function of the rotational to gravitational energy#$2MNRAS 000, 000–0
c 0000 RAS,⃝x=exp[−a]1
2gravitational2
and magnetic
toenergyx = 2 1 − ln(ab/2) + ln(a/2 −$ln(ab/2)) %の変更。x → 0 で支配的な項のみを取り出すと、
b
a
a
GM
ρ
r
i
!
"
#
−1
r 1= ri exp −8π21B1ln xB 2 /ri。[] の第一項までが、角運動量が保存したときの円盤半径に対応
0i
′−1
′
Disk
radius[ji − j(x)] =x dx = −(71)
2$2Ωi ri GMr ρi /r2i #√8π x8π
円盤半径はjji = GM2r ri x を代入して得る。
2!
"r=
1 − ln(ab/2)
2 + %ln(a/2 − ln(ab/2))
$
!
"
j
B
1/2
i
2
GM
a
a
r
GM0ir /ri ln x 1/2
1B0i (72)
ji − j(x)=−1 − GMr ρi /r2i
x =−
ln x
8π
方、x → 1 のとき、 radius without(Ωi ri ) magnetic ﬁeld
8π GMr ρi /ri

!

#
$
1
1 f$
ρ%i1/2
$ !
"%−1
2
2
1−
(1 − x)GM
1 [ji − j(x)]
B0i =
r /ri
2
a=
(73)
Ωi ri GMr ρi /r
4π bf =−1ρ̄(ri )i
2
8π GMr ρi /ri(Ωi ri )= 1 − ϵ とする。このとき
角運動量が0 になる半径は、
B0i2

"−1

Anglar Momentum in Envelope-1

j [km pc s ]Ωi = 8.1 x 10-14 s-1, B0i = 14.3 µG, n0i = 105 cm-3
10

-1

10

-2

t = 2x105 yr (Mps~0.5Msun)

jenv
jkep
jini

Kepler rotation
without magnetic ﬁeld

10-3
10

result of the model

-4

10

100

1000

10000

r [au]

Model predict that disk radius is ~300 au
These results should be compared with simulations.

Comparison with a simulation
disk radius ~100 au(model prediction is ~300 au)

rdisk ji2 ri4Strongly depends on
initial radius ri
Farther comparison withthe simulation and updateof the model is requiredSummary•

Angular momentum of cloud cores is important for protoplanetary disk
formation, so that it is also important for star and planet formation.

•

Without magnetic ﬁeld, angular momentum of the infalling envelope isconserved and ﬂat j proﬁle is formed.

•

Magnetic ﬁeld transfers the angular momentum in the envelope and willstrongly aﬀect the disk formation.

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We develop the analytic model for the infalling envelope with magnetic ﬁeld
and investigate the time evolution of the angular momentum in the envelope.

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Compered with the numerical simulation, the model overestimate the disk
radius. This may caused by the assumption of the spherical collapse in themodel.

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Farther comparison with the simulations and update of the model is required.