Recurrent Planet Formation and Intermittent Protostellar Outflows Induced by Episodic Mass Accretion

1.5 MB·English

by  Masahiro N. Machida

Recurrent Planet Formation and Intermittent Protostellar
Outflows Induced by Episodic Mass Accretion
Masahiro N. Machida1, Shu-ichiro Inutsuka2, and Tomoaki Matsumoto3
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1
ABSTRACT
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2

n
a The formation and evolution of a circumstellar disk in magnetized cloud cores
J
is investigated from prestellar core stage until ∼ 104yr after protostar formation.
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1 In the circumstellar disk, fragmentation first occurs due to gravitational insta-

bility in a magnetically inactive region, and substellar-mass objects appear. The
]
R
substellar-mass objects lose their orbital angular momenta by gravitational in-
S
teraction with the massive circumstellar disk and finally fall onto the protostar.
.
h
After this fall, the circumstellar disk increases its mass by mass accretion and
p
- again induces fragmentation. The formation and falling of substellar-mass ob-
o
r jects are repeated in the circumstellar disk until the end of the main accretion
t
s
a phase. In this process, the mass of fragments remain small, because the circum-
[
stellar disk loses its mass by fragmentation and subsequent falling of fragments

1
before it becomes very massive. In addition, when fragments orbit near the
v
7 protostar, they disturb the inner disk region and promote mass accretion onto
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the protostar. The orbital motion of substellar-mass objects clearly synchronizes
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1 with the time variation of the accretion luminosity of the protostar. Moreover,
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1 as the objects fall, the protostar shows a strong brightening for a short duration.
0
1 The intermittent protostellar outflows are also driven by the circumstellar disk
1
whose magnetic field lines are highly tangled owing to the orbital motion of frag-
:
v
ments. The time-variable protostellar luminosity and intermittent outflows may
i
X
be a clue for detecting planetary-mass objects in the circumstellar disk.
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a
Subject headings: accretion, accretion disks: ISM: clouds—stars: formation—
stars: low-mass, brown dwarfs: planetary systems: planetary disks
1National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan;
[email protected]
2Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602;
[email protected]
3Faculty of Humanity and Environment, Hosei University, Fujimi, Chiyoda-ku, Tokyo 102-8160, Japan;
[email protected]
– 2 –
1. Introduction
The observation of many star-forming regions has shown that stars form in molecular
cloud cores. A star is born after a long journey through the gravitational collapse of a cloud
core. Thegasbeginstocollapseinadensepartofthecloud(i.e., thecloudcore)andincreases
its density and temperature as the cloud core collapses further. When the collapsing gas
reaches a sufficiently high density (∼ 1021cm−3), the collapse stops and a protostar having
almost a Jovian mass is born (Larson 1969). Then, the protostar increases its mass through
mass accretion and finally evolves into a main-sequence star (or main-sequence phase).
Thestarformationprocesscanbedividedintotwophases: theearlyprotostellarcollapse
phase and later accretion phase of the star formation. The early phase, also called the gas
collapsing phase, is defined as the period before protostar formation after gas collapse is
initiated in the cloud core. Thus, during this phase, the gas continues to collapse. On the
other hand, the later phase is defined as the period before the main-sequence phase after
the gas collapse stops and a protostar appears in the collapsing cloud core. In particular,
during the later phase, the period when the protostar significantly increases its mass by mass
accretion is called “the main accretion phase.” Since the star acquires almost all its mass
in this main accretion phase, the final stellar mass is determined in this phase. Protostars
during the main accretion phase are observed (or defined) as Class 0 or I objects by spectral
energy distribution (SED). The observation of star-forming regions also has shown that
almost all Class 0 or I objects drive a protostellar outflow. Because a certain fraction
of mass and angular momentum are ejected by the protostellar outflows, they are closely
related to the rate of mass accretion onto the protostar and final stellar mass. Moreover,
the infalling gas with an angular momentum forms the circumstellar disk during the main
accretion phase. The evolution of the circumstellar disk is related to the mass accretion and
outflow rates. In addition, the circumstellar disk is the site of planet formation. Therefore,
understanding the evolution of the circumstellar disk during the main accretion phase is
essential to understanding both the star and planet formation processes.
Since the protostar and circumstellar disk during the main accretion phase are veiled by
the infalling envelope, it is difficult to observe them. Recently, however, the Spitzer Space
Telescope (SST) has been unveiling this phase. Enoch et al. (2009a) discovered a massive
circumstellar disk of ∼ 1M comparable to a central protostar around a Class 0 object,
⊙
indicating that (i) the disk already exists in the main accretion phase, and (ii) the disk mass
is significantly larger than the theoretical prediction. Evans et al. (2009) observed several
star forming regions and identified several Class 0 objects using the SST. They showed that
the bolometric luminosity of the objects is considerably dimmer than classical theoretical
predictions, and has a dispersion over 2-3 orders of magnitude. They pointed out that recent
– 3 –
observations aggravate the “luminosity problem” (the problem that the accretion luminosity
of protostars is lower than theoretical predictions, see Kenyon et al. 1990), and concluded
that non-steady accretion is inevitably required to explain the observational results. The
non-steady accretion may be attainable when the circumstellar disk is sufficiently massive
and causes the gravitational instability (Durisen et al. 2007).
Another recent development in observation of star and planet formation is direct images
ofexo-planets (Kalas et al. 2008; Marois et al.2008; Thalmann et al. 2009), inwhich planets
arelocatedat> 10AUfromthecentral star. However, intheframework ofthecoreaccretion
∼
scenario (Hayashi et al. 1985), there is less likehood of planets forming in a region that is so
remote from the central star. Alternatively, the gravitational instability scenario (Cameron
1978) may explain the formation of such planets. In summary, recent observations seem to
indicate that the protostars have a considerably massive disk unlike what was previously
believed.
Although it seems that recent observational progress is unveiling problems for the early
evolution stage of star formation, we cannot directly observe the circumstellar disk and pro-
tostarinthemainaccretionphasebecause theyareembedded inthedenseinfalling envelope.
Thus, theoretical study is necessary to understand the properties of the circumstellar disk
and protostar. In particular, multi-dimensional simulations are necessary to investigate the
evolution of the circumstellar disk, protostellar outflow, and so on.
The evolution of the collapsing gas cloud core from the protostellar core stage (i.e.,
the gas collapsing phase) until protostar formation has been well investigated using multi-
dimensionalsimulations(e.g.,Bate1998;Tomisaka2002;Whitehouse & Bate2006;Stamatellos et al.
2007; Banerjee & Pudritz 2006; Machida et al. 2006, 2007, 2008a,b, 2009b). On the other
hand, only a few studies have focused on the main accretion phase immediately following
the prestellar core stage, because it is difficult to calculate long-term evolution in the main
accretion phase with sufficient spatial resolution. In unmagnetized cloud cores, the forma-
tion and evolution of the circumstellar disk in the main accretion phase through the gas
collapsing phase were investigated by Walch et al. (2009a,b), Vorobyov & Basu (2010), and
Machida et al. (2010). They found that the circumstellar disk is considerably massive to in-
duce fragmentation or the gravitational instability that is related to a non-steady accretion
flow onto the protostar.
In reality, however, since molecular clouds are strongly magnetized (Crutcher 1999), the
magnetic field may play an important role in the evolution of the circumstellar disk during
the main accretion phase. Vorobyov & Basu (2006, 2007) investigated the evolution in two
dimensions of the circumstellar disk in a magnetized cloud core and showed the non-steady
accretion onto the central protostar. In three dimensions, the formation and evolution of the
– 4 –
circumstellar disk from prestellar core stage were investigated only by Inutsuka et al. (2009),
in which they showed fragmentation and possible planet formation in the magnetically in-
active region of the circumstellar disk during the main accretion phase. They also indicated
non-steady mass accretion onto the protostar owing to the gravitational instability of the
circumstellar disk. However, this study only calculated the evolution of the circumstellar
disk about ∼ 1000yr after protostar formation.
In this study, in a setting similar to Inutsuka et al. (2009), we investigate the evolution
of the circumstellar disk for ∼ 104yr, which is ∼ 10 times longer than the previous study. In
addition to the model adopted by Inutsuka et al. (2009), we newly calculate the evolution
of the circumstellar disk formed in a relatively stable initial cloud core. In both models, we
compare the mass accretion rate, properties of protostellar outflow, and the fragmentation
condition. The structure of the paper is as follows. The framework of our models and the
numerical method are given in §2. The numerical results are presented in §3. We discuss the
fragmentation condition of the circumstellar disk andits implication forthe planet formation
in §4, and we summarize our results in §5.
2. Model and Numerical Method
2.1. Basic Equations
To study the formation and evolution of a circumstellar disk in a magnetized molecular
cloud core, we solve the three-dimensional resistive MHD equations, including self-gravity:
∂ρ
+∇·(ρv) = 0, (1)
∂t
∂v 1
ρ +ρ(v ·∇)v = −∇P − B ×(∇×B)−ρ∇φ, (2)
∂t 4π
∂B
= ∇×(v ×B)+η ∇2B, (3)
OD
∂t
∇2φ = 4πGρ, (4)
where ρ, v, P, B, η , and φ denote density, velocity, pressure, magnetic flux density,
OD
resistivity, and gravitational potential, respectively. In addition, we adopted the hyperbolic
divergence cleaning method of Dedner et al. (2002) to obtain divergence-tree magnetic field
(∇·B = 0). With this method, no magnetic monopoles appear throughout calculation (see
also Machida et al. 2005a; Matsumoto 2007).
To mimic the temperature evolution calculated by Masunaga & Inutsuka (2000), we
adopt the piece-wise polytropic equation of state (see Vorobyov & Basu 2006; Machida et al.
– 5 –
2007) as
2/5
ρ
P = c2 ρ 1+ , (5)
s,0 ρ
" (cid:18) c(cid:19) #
where c = 190ms−1 and ρ = 3.84×10−14g cm−3 (n = 1010cm−3). With equation (5),
s,0 c c
the gas behaves isothermally for n < 1010cm−3 and adiabatically for n > 1010cm−3. For
∼ ∼
a realistic evolution of the magnetic field in the circumstellar disk, we adopt the resistivity
(η ) as the fiducial value in Machida et al. (2007), in which the Ohmic dissipation becomes
OD
effective for 1011cm−3 < n < 1015cm−3 (for details, see Eq. [9] and [10], and Fig. 1 of
∼ ∼
Machida et al. 2007).
2.2. Initial Setting and Numerical Method
As the initial state, we take a spherical cloud core with a critical Bonnor–Ebert (BE)
density profile, in which a uniform density (ρ ≃ 0.07ρ ) is adopted outside the sphere
amb c
(r > R ). For the BE density profile, we adopt the central density as n = 106cm−3
c c
and isothermal temperature as T = 10K. For these parameters, the critical BE radius is
R = 4.7 × 103AU. The gravitational force is ignored outside the host cloud (r > R ) to
c c
mimic a stationary interstellar medium. In addition, we prohibit the gas inflow at r = R to
c
suppress mass input from the interstellar medium into the gravitationally collapsing cloud
core. Thus, only the gas inside r < R collapses to form the circumstellar disk and protostar.
c
Note that although the protostellar outflow driven by the circumstellar disk propagates into
the region of r > R and disturbs the interstellar medium over time, we can safely calculate
c
the mass accretion process onto the circumstellar disk because the outflow propagating into
the interstellar medium does not affect the inner cloud core region (r < R ). In this study,
c
we call this initial spherical cloud with Bonnor–Ebert density profile ‘the cloud core’ that
has a radius of R and mass of M .
c cl
Since thecritical BEsphere isinanequilibrium state, we increase thedensity byafactor
of f to promote the contraction, where f is the density enhancement factor that represents
the stability of the initial cloud core. An initial cloud core with larger f is more unstable
against gravity. In general, the stability of the cloud core is represented by a parameter
α (≡ E /E ) that is the ratio of thermal (E ) to gravitational (E ) energy. As
0 th grav th grav
shown in Matsumoto & Hanawa (2003), when the BE density profile is adopted, the density
enhancement factor is related to the parameter α as
0
0.84
α = . (6)
0
f
We constructed two models with different f (= 1.68 and 2.8). In this paper, we call the
– 6 –
model having f = 1.68 (α = 0.5) “model A05,” and the model having f = 2.8 (α = 0.3)
“model A03.” At the initial state, model A03 is more unstable than model A05.
In both models, the initial cloud core rotates rigidly around the z-axis in the region of
r < R , while the uniform magnetic field parallel to the z-axis (or rotation axis) is adopted in
c
the whole computational domain. In addition, we adopt that the angular velocity decreases
in proportional to ∝ exp(−r2) outside the host cloud (r > R ). Both models have the same
c
angularvelocity withΩ = 1.1×10−13s−1 andthesamemagneticfield, B = 37µG.Themass
0 0
inside r < R for each model is 0.8M (model A05) and 1.3M (model A03). Model names
c ⊙ ⊙
and initial values are summarized in Table 1. The initially magnetized, rotating cloud core
is also specified by the non-dimensional parameters β and γ , where β (≡ E /|E |) and
0 0 0 rot grav
γ (≡ E /|E |) is the ratio of the rotational and magnetic energy to the gravitational
0 mag grav
energy inside the initial cloud core. Model A03 has β = 5 × 10−3 and γ = 0.05, while
0 0
model A05 β = 6 × 10−3 and γ = 0.07. In addition, the mass-to-flux ratio M/Φ is also
0 0
used to specify the initial cloud, where M and Φ are the mass and magnetic flux of the initial
cloud core. There exists a critical value of M/Φ below which a cloud is supported against
the gravity by the magnetic field. For a cloud with uniform density, Mouschovias & Spitzer
(1976) derived a critical mass-to-flux ratio
1/2
M ζ 5
= , (7)
Φ 3π G
(cid:18) (cid:19)cri (cid:18) (cid:19)
where the constant ζ = 0.53. We define the mass-to-flux ratio normalized by the critical
value as λ. In our setting, model A03 has λ = 9, while model A05 has λ = 5.6.
We add m = 2-mode non-axisymmetric density perturbation to the initial cloud core.
Then, the density profile of the cloud core is described as
ρ (r)(1+δ )f for r < R ,
BE ρ c
ρ(r) = (8)
ρ (R )(1+δ )f for r ≥ R ,
(cid:26) BE c ρ c
where ρ (r) is the density distribution of the critical BE sphere, and δ is the axisymmetric
BE ρ
density perturbation. For the m = 2-mode, we chose
δ = A (r/R )2cos2φ, (9)
ρ φ c
where A (=0.01) represents the amplitude of the perturbation. The radial dependence is
φ
chosen so thatthe density perturbationremains regular atthe origin(r = 0) at onetime-step
after the initial stage. This perturbation ensures that the center of gravity is always located
at the origin.
In the collapsing cloud core, we assume the protostar formation to occur when the
number density exceeds n > 1013cm−3 atthecenter. Tomodel theprotostar, weadoptasink
– 7 –
around the center of the computational domain. In the region r < r = 1AU, gas having
sink
a number density of n > 1013cm−3 is removed from the computational domain and added to
the protostar as gravity in each timestep (for details, see Machida et al. 2009a, 2010). This
treatment of the sink makes it possible to calculate the evolution of the collapsing cloud
core and circumstellar disk for a longer duration. In addition, inside the sink, the magnetic
flux is removed by Ohmic dissipation, because such a region has the magnetic Reynolds Re
number exceeding unity Re > 1 (for details, see Machida et al. 2007).
To calculate on a large spatial scale, the nested grid method is adopted (for details,
see Machida et al. 2005a,b). Each level of a rectangular grid has the same number of cells
(128 ×128 × 32). The calculation is first performed with five grid levels (l = 1 − 5). The
box size of the coarsest grid l = 1 is chosen to be 25R . Thus, grid of l = 1 has a box
c
size of ∼ 1.5×105AU. A new finer grid is generated before the Jeans condition is violated
(Truelove et al. 1997). The maximum level of grids is l = 12 that has a box size of 74AU
max
and a cell width of 0.58AU.
We adopted a fixed boundary condition on the outermost grid boundary. The uniform
density (ρ ) and magnetic field (B = 0, B = 0, B = B ) and zero fluid velocity
amb x y z 0
(v = v = v = 0) are imposed on l = 1 grid boundary at each timestep. Such boundary
x y z
conditionhardlyaffectstheevolution ofthecloudcore, because thegravitationallycollapsing
cloud core with a radius of 4.7×103AU (r = R ) is embedded in a large simulation box with
c
a size of ∼ 3×105AU, inside which the static interstellar medium is fulfilled in the region
of r > R . In our setting, the Alfv´en speed in the interstellar medium (r > r ) for model
c BE
A03 is v = 0.15kms−1. Thus, by reaching the Alfv´en wave to the computational boundary
A
from the center of the cloud core, it takes 4.7×106yr. Since we stopped our calculation in
∼ 104yr after the calculation begins, the Alfv´en wave generated at the center of the cloud
core (or the computational boundary) never reaches the computational boundary (or the
center of the cloud core).
3. Results
We calculated the formation and evolution of the circumstellar disk in magnetized col-
lapsing cloud cores. First, we simply outline the formation and evolution of the circumstellar
disk. Before protostar formation, the first (adiabatic) core forms in the collapsing cloud core
witha scale of∼ 10AU(Larson1969;Masunaga & Inutsuka 2000; Saigo & Tomisaka 2006).
At its formation, the first core has a thick disk-like structure and is mainly supported by the
thermal pressure gradient force. After protostar formation, the first core becomes thin be-
causethecentrifugal forcedominatesthethermalpressure gradientforce(Walch et al.2009a;
– 8 –
Machida et al. 2010). During the main accretion phase, the first core grows and extends to
a large extent (∼ 100−1000AU). Inutsuka et al. (2009) and Machida et al. (2010) pointed
out that the circumstellar disk originates from the first core, and is formed before protostar
formation (see also Bate 1998, 2010). Their calculations showed that the first core gradually
grows to become the circumstellar disk in the main accretion phase. They also pointed out
that, reflecting the thermal history of the collapsing cloud core, the circumstellar disk is in-
evitably more massive than the protostar in the early part of the main accretion phase. The
formation of the first core (or the circumstellar disk) before protostar formation has been
investigated in many past studies (see review by Bodenheimer et al. 2000; Goodwin et al.
2007). Thus, although we began the calculation from the prestellar core stage, we mainly
focus on the evolution of the circumstellar disk in the main accretion phase in this paper.
3.1. Recurrent Fragmentation and Planet Formation
As described in §2, we constructed two models (models A03 and A05) with different
initial stabilities (i.e., different α ). In the two models, the protostar forms t ≃ 2.50×104yr
0
(model A03) and t ≃ 5.48× 104yr (model A05), respectively, after the calculation begins.
Thus, in the collapsing cloud core, the protostar for model A03 forms for a shorter duration
thanthatformodelA05. ThisisbecausetheinitialcloudcoreformodelA03ismoreunstable
than that for model A05 and begins to collapse according to the self-similar solution (Larson
1969) in a shorter time after the calculation begins (Machida et al. 2008a). In both models,
we calculated the evolution of the circumstellar disk for ∼ 104yr after protostar formation.
First, we describe the evolution of the circumstellar disk for model A03 that shows
the recurrent fragmentation and formation of substellar-mass object in the main accretion
phase. Figure 1 shows the evolution of the circumstellar disk after protostar formation for
model A03, in which the density distributions around the protostar on the equatorial plane
are plotted. The box sizes in the upper panels are ∼ 80AU, while those in the middle
and lower panels are ∼ 320AU. Figure 1a shows the structure of the circumstellar disk
t ≃ 450yr after protostar formation, in which the red region (n > 1011cm−3) corresponds
c ∼
to the circumstellar disk. Note that we describe the elapsed time after protostar formation
with a notation of ‘t ,’ which differs from the elapsed time ‘t’ after the calculation begins
c
(or the cloud begins to collapse). The size of the circumstellar disk increases with time and
extendsupto∼285AUbytheendofthecalculation. Inthecircumstellar disk, fragmentation
occurs due to gravitational instability t ≃ 630yr after protostar formation (for details, see
c
Inutsuka et al. 2009; Machida et al. 2010). Two ambiguous clumps appear in Figure 1b and
c, while two clear fragments are seen in Figure 1d. Fragmentation occurs ∼ 6.8AU away
– 9 –
from the protostar. Then, fragments orbit around the protostar with an orbital separation
of ∼ 2−38AU (Figs. 1c-e) and finally fall onto the protostar at t ≃ 3760yr (Figs. 1e and f).
c
In this section, we call the fragment appeared in the circumstellar disk the planet. Note that
in reality, since some fragments that appeared in this calculation exceeded the deuterium-
burning limit (about 13 Jupiter mass), they are not planets in the complete sense. Note
also that we define, for convenience, the objects born in the circumstellar disk as planets to
characterize them as a whole: we discuss their mass in §4.2.
Figure 2 shows the structure of the circumstellar disk and protostellar outflow at almost
the same epoch as Figure 1c in three-dimensions. The outflow driven by the first core in
the collapsing cloud core has been studied by many authors (e.g., Matsumoto & Tomisaka
2004; Machida et al. 2004; Machida et al. 2005a; Hennebelle & Fromang 2008a). Even in
the main accretion phase (i.e., even after protostar formation), the circumstellar disk that
originates from the first core continues to drive outflow (Tomisaka 2002; Banerjee & Pudritz
2006; Machida et al. 2007, 2009a). In this model, the outflow driven by the circumstellar
disk extends up to ∼516AU by the end of the calculation. As seen in Figure 2, the outflow
is driven only by the outer disk region, while the outflow is not driven by the inner region of
the circumstellar disk where fragmentation occurs and planets appear. This is because the
inner disk region has a density of n > 1011 −1012cm−3 and the magnetic field is dissipated
∼
by Ohmic dissipation. As shown in Nakano et al. (2002) and Machida et al. (2007), when
the number density exceeds n > 1011−1012cm−3, Ohmic dissipation becomes effective owing
∼
to the extremely low ionization degree. Note that the ambipolar diffusion can be effective in
the range of n < 1011 −1012cm−3 (see, §4.3). Fragmentation is suppressed by the magnetic
field because magnetic effects such as magnetic braking and outflows effectively transfer
the angular momentum that promotes fragmentation (Machida et al. 2005b; Machida et al.
2008a; Hennebelle & Teyssier 2008b). Thus, a weaker field promotes fragmentation in the
circumstellar disk. As a result, the inner disk region has a considerable weak magnetic field
and fragmentation occurs without outflow, while the outer disk region has a strong field and
no fragmentation occurs with outflow.
In Figure 2, yellow lines are magnetic field lines. In the figure, only magnetic field lines
that threaded planets are plotted to stress the planet’s spin or orbital motion. The magnetic
field lines just above each planet are strongly twisted by the spin motion of the planet. In
the midair (or on a large scale), the magnetic field lines originating from each planet are
twisted together, reflecting the orbital motion of the planets. Although we only plotted
magnetic field lines near the planet to understand the planet’s motion, the magnetic field
(or Lorentz force) in this region (or inner disk region) is extremely weak (β ≫ 10, where
p
β is the plasma beta and different from initial ratio of the rotational to the gravitational
p
energy, β ) because Ohmic dissipation is effective. Thus, near the planets, the magnetic field
0
– 10 –
lines are passively moved, and they hardly affect the dynamical evolution of both the planet
and circumstellar disk. However, planets perturb the outer disk region that is connected
to magnetic field lines driving outflow (i.e., the magnetic field lines in the outer disk region
where Ohmic dissipation is not effective). As a result, a highly time-variable outflow appears
in this model. We will discuss an intermittent outflow in §3.7.
Figure 1e shows the structure of the circumstellar disk just before the planets fall onto
the protostar, while Figure 1f shows the disk just after the fall. After the first-generation
planets disappear, fragmentation occurs again in the circumstellar disk and the second-
generation planets appear t ≃ 3990yr after protostar formation as shown in Figure 1g.
c
However, these planets also fall onto the protostar ∼ 680yr after their formation. The
formation and falling of the planets are repeated several times in the circumstellar disk.
By the end of the calculation, 18 planets appeared and 16 planets fell onto the protostar.
Figure 3 shows eighth- and ninth-generation planets. The eighth-generation planets fell onto
the protostar, while the ninth-generation planets survived until the end of the calculation.
The orbital radius r , time t at planet formation, planet’s falling epoch t , lifetime t −t ,
i i e e i
and mass at its formation M are summarized in Table 2.
pl
3.2. Orbital Trajectories of Planets
Figure 4 shows the orbital evolution of planets against time after protostar formation
t . When multiple planets appear as seen in Figures 1 and 3, only the orbital radius of the
c
most massive planet is plotted. The figure indicates that the first-generation planets form at
∼ 6.8AU from the protostar at t ≃ 795yr, and shrink their orbit and fall onto the protostar
c
after ∼ 10 orbital rotations. The second-generation planets appear at ∼ 46AU and fall
onto the protostar after ∼ 2 −3 orbital rotations. This formation and falling of planets is
repeated, until the ninth-generation planet is formed.
Figure 5 shows the trajectory for planets for each generation. The figure indicates that
planets shrink their orbits with a certain amount of eccentricity. Figures 4 and 5 show
that planets are born at ∼ 6−50AU from the protostar and orbit for ∼ 500−1000yr (or
2−10 orbital rotations) around the protostar. In addition, these figures show that the first-
generation planets are formed at a relatively small orbital radius (∼ 6.5AU) and orbit for
∼ 10 times around the protostar, while other generations of planets are formed at a larger
orbital radii (∼ 20−50AU) and orbits only for 2−3 times. This is because the circumstellar
disk increases its size and mass with time. As shown in Figure 1, the circumstellar disk has
a size of ∼ 20−30AU when the first-generation planet appears, while the disk size reaches
∼ 200AU just before the second-generation planet appears. As a result, the magnetically