【Star Formation News Letter速報ゼミ】

20170728:
古屋 玲 (徳島⼤学・教養教育院)☆★☆★☆★☆★☆★☆★☆★☆★☆★☆★☆★☆★
【星形成ゼミ】★日時: 7月28日 (金)☆場所: 理論部セミナー室 (中央棟南３階)★時間: 10:30-12:00☆論文速報 SFN #294
★古屋さん (11-15本目)☆徳田さん (16-20本目)★資料置き場: http://th.nao.ac.jp/MEMBER/kataoka/star/?2017%C7%AF%C5%D9%C1%B0%B4%FC★☆★☆★☆★☆★☆★☆★☆★☆★☆★☆★☆★☆

Diﬀerent dust and gas radial extents in protoplanetary disks: consistent models of grain
growth and CO emission
1

Stefano Facchini , Til Birnstiel

2,3

1

1,4

, Simon Bruderer and Ewine F. van Dishoeck

1.1.

1

Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany
Max-Planck-Institut für Astronomie, Königstuhl, D-69117 Heidelberg, Germany
3
University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich,Germany
4
Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, The Netherlands
2

DALI_NEW

facchini@mpe.mpg.d

E-mail contact: facchini at mpe.mpg.de【背景と動機】
ALMA observations of protoplanetary disks confirm earlier indications that there is a clear diﬀerence between the dust
and gas radial extents. The origin of this diﬀerence is still debated, with both radial drift of the dust and optical depth• 原始惑星系円盤の進化を考えるうえで鍵になる物理量：ガス⾯密度の動径分布Σ®と円盤外径R。ミリ波⼲渉計観測か
eﬀects suggested in the literature. In thermo-chemical models, the dust properties are usually prescribed by simple
parametrisations.
In this work, the feedback of more realistic dust particle distributions onto the gas chemistry
andら、CO輝線とダスト熱放射をトレーサーに選んだ場合、⼀般にR(12CO)> R(dust)が1990年代から知られていた。両molecular emissivity is investigated, with a particular focus on CO isotopologues. The radial dust grain size distribution
者の光学的厚みの差に起因と説明されてきた。
is determinedusing dust evolution models that include growth, fragmentation and radial drift for a given static gas
density structure. The vertical settling of dust particles is computed in steady-state. A new version of the code
• 近年では、ダスト成⻑および⼤きなダストの円盤内側への移動、円盤⾚道⾯への沈着で説明されることが多い。
DALI
is used to take into account how dust surface area and density influence the disk thermal structure, molecularabundances and excitation. Synthetic images of both continuum thermal emission and low J CO isotopologues lines• そこで「現実的な化学進化」や外部UV輻射場を考慮してモデルの精密化がさらに必要。
are produced. The diﬀerence of dust and gas radial sizes is largely due to diﬀerences in the optical depth of CO lines
and millimeter continuum, without the need to invoke radial drift. The eﬀect of radial drift is primarily visible in the
sharp outer edge of the continuum intensity profile. The gas outer radius probed by 12 CO emission can easily diﬀer
by a factor of ∼ 2 between the models for a turbulent α ranging between 10−4 and 10−2 , with the ratio of the CO and【本研究と著者らの先⾏研究】
out
out
mm radius RCO/Rmm
increasing with turbulence. Grain growth and settling concur in thermally decoupling the gas
andcomponents, due to the low collision DALI
rate with(Dustlarge grains.
a result, the gas can be much colder than• dust図１：著者らによるstandard
And AsLInes)コードと本研究(＝オレンジで囲んだ四⾓形部分)の関係
the dust at intermediate heights,
reducing
CO excitation
emission,for disks
low turbulence values. Also,
S. Facchini
et al.:theDi↵erent
dust and gasandradialextents inespecially
protoplanetary
due to disk mid-plane shadowing, a second CO thermal desorption (rather than photodesorption) front can occur inthe warmer outer mid-plane disk. The models are compared to ALMA observations of HD 163296 as a test case. In
order to reproduce the observed CO snowline of the system, a binding energy for CO typical of ice mixtures, with
Eb ≥ 1100
K,densityneeds tobe used rather than the lower
value.
The diﬀerence between
observed
gas and dustGas
structure
Radialpure
grainCOsize
distribution
Dust density
distribution
extent is2D
largelydueintoR opticalbut Uniform
radial drift
and grainalso aﬀect
thegas and dust
structureand z depth
dust-to-gas
ratiosize evolution
Verticalsettling?, vfrageﬀects,, ?gr?, ?gremission inStellar
subtleparameters
ways. In order to properly infer fundamental
ofal.
disks,
such as(Birnstiel etquantities
al. 2015) of the gaseous component(Birnstiel et2010)
disk outer radii and gas surface density profiles, simultaneous modelling of both dust and gas observations including
dust evolution is needed.
Accepted by A&A
https://arxiv.org/pdf/1705.06235
abudances,
Tgas GW Ori
Dust temperature
Millimeter
ObservationsChemicalof the disk
aroundObservables
Thermal balanceMonte-Carlo RT

Average grain size

47
8
M. Fang1,2Ray, A.tracing
Sicilia-Aguilar3,1Grain, D. Wilner, Y. Wang5,6 , V. Roccatagliata, andetJ.Z.
Wang9
size dependence
New opacities , D. Fedele(Vasyuninal. 2011)
1(Bruderer 2013)(Bruderer
et al. Autnoma
2012, 2013)(Bruderer
et al. 2012)
Departamento de Fsica Teórica,
Universidad
de Madrid,
Cantoblanco28049, Madrid, Spain
2
Department of Astronomy, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
3
SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
4Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
5 Fig. 1. Diagram of the method used in this paper. The blue box indicates the input parameters and the green box indicates the output observables.
Max Planck
Institute
Astronomy,Königstuhl
Boxeswith orange
bordersfor
show
new modules,
compared to17,
the69117,
standardHeidelberg,
DALI code. Germany
Orange text underlines modifications in previous DALI
6
Purple The
MountainObservatory
and needed
Key Laboratory
Radio
Astronomy, Chinese Academy of Sciences, 2 West
modules.
red text lists
input parameters
to compute theofdust
distribution.
Beijing Road, 210008 Nanjing, China

Diﬀerent dust and gas radial extents in protoplanetary disks: consistent models of grain
growth and CO emission
1

Stefano Facchini , Til Birnstiel

2,3

1

1,4

, Simon Bruderer and Ewine F. van Dishoeck

1.2.

1

Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany【計算結果】
Max-Planck-Institut für Astronomie, Königstuhl, D-69117 Heidelberg, Germany

2

University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich,•
ダストの密度分布、グレインサイズ（平均値）の動径分布[図3]、ダスト温度の動径分布[図5]、ダストサイズ（-->ダス
Germany
4
LeidenObservatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, The Netherlands
トサイズ成⻑を反映）の動径分布と⾚道⾯への沈着、CO柱密度や存在⽐の動径分布[図8,9]など。
3

E-mail contact: facchini at mpe.mpg.de• HD163296にモデルを適⽤。

Accepted by A&A
https://arxiv.org/pdf/1705.06235

Millimeter Observations of the disk around GW Ori

COから求めた円盤外径とダスト熱輻射からの外径の⽐A&A disks
proofs: manuscript
ms_le indications that there is a clear diﬀerence between the dust
ALMA observations of protoplanetaryconfirmno.
earlierand gas radial extents. The origin of this diﬀerence is still debated, with both radial drift of the dust and optical depth【主な議論と「まとめ」の抜粋】
eﬀectssuggested in the literature. In thermo-chemical models, the dust properties are usually prescribed by simple
parametrisations. In this work, the feedback of more realistic dust particle distributions onto the gas chemistry and• ダストのCOによる円盤サイズの差異は、結局、両者の光学的厚さの差異で説明可能；ダストの動径⽅向移動(radialmolecularemissivity is investigated, with a particular focus on CO isotopologues. The radial dust grain size distribution
is determinedusing dust evolution models that include growth, fragmentation and radial drift for a given static gas
drift)を必要としない。なお、ダストの動径⽅向移動の効果はサブミリ撮像で明瞭なedgeとして検出されるはず。
density structure. The vertical settling of dust particles is computed in steady-state. A new version of the code
DALI
is used to take into account how dust surface area and density influence the disk thermal structure, molecular• 円盤ガスの乱流の程度が⼩さければ「⼤きなダスト」は成⻑＆⾚道⾯へ沈着：これらの過程はダストとCO強度動径⽅向
abundances and excitation. Synthetic images of both continuum thermal emission and low J CO isotopologues lines
are分布を⽐較すればモデルから制限できる（図17；右下）。
produced. The diﬀerence of dust and gas radial sizes is largely due to diﬀerences in the optical depth of CO lines
andmillimeter
continuum,
withouttheinvokeradialeﬀect
oftheradialdrift
is primarily visible in theFig.
8. Leftand central panel:column densityof CO gasand need
CO ice to(JCO)of all models.
Wedrift.
note theThedi↵erentscale ony-axis in
the twoA&A proofs: man
3
12panels:N(JCO)
isalways
much
lower
than
N(CO)
in
the
gas
phase
at
all
radii,
even
for
↵
=
10
.
Right
panel:
abundanceofCO(solidlines)
andsharp outer edge of the continuum intensity profile. The gas outer radius probed by CO emission can easily diﬀer
CO ice (dashed COとアイスの存在⽐分布のα依存性
lines) along the disk mid-plane.
；αは円盤内の粘性の最⼤最⼩値の⽐(Hartmann 98)；α値が⼩さいと乱流の程度は低い。
by a factor of ∼ 2 between the models for a turbulent α ranging between 10−4 and 10−2 , with the ratio of the CO andoutout
mm radius RCO/Rmm
increasing with turbulence. Grain growth and settling concur in thermally decoupling the gas
and dust components, due to the low collision rate with large grains. As a result, the gas can be much colder thanthe dust at intermediate heights, reducing the CO excitation and emission, especially for low turbulence values. Also,
due to disk mid-plane shadowing, a second CO thermal desorption (rather than photodesorption) front can occur inthe warmer outer mid-plane disk. The models are compared to ALMA observations of HD 163296 as a test case. In
乱流的な円盤ではCOで測った半径はorder to reproduce the observed CO snowline of the system, a binding energy for CO typical of ice mixtures, with
ダストで測った半径の数倍に!
Eb ≥ 1100 K, needs to be used rather than the lower pure CO value. The diﬀerence between observed gas and dust
extent is largely due to optical depth eﬀects, but radial drift and grain size evolution also aﬀect the gas and dust
emission in subtle ways. In order to properly infer fundamental quantities of the gaseous component of disks, such as
disk outer radii and gas surface density profiles, simultaneous modelling of both dust and gas observations including
dust evolution is needed.

M. Fang1,2 , A. Sicilia-Aguilar3,1, D. Wilner4 , Y. Wang5,6 , V. Roccatagliata7 , D. Fedele8 , and J.Z. Wang91

Departamento de Fsica Teórica, Universidad Autnoma de Madrid, Cantoblanco 28049, Madrid, Spain Fig. 17. Ratio Rout /Routmm derived for the models using the disk parametriCO
Department of Astronomy, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA sation of de Gregorio-Monsalvo et al. (2013), with a 0.5200 synthetic
3
SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS,UKbeam.
The quantity Rout
CO is derived assuming a dynamic range of 30 for
4
12
Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
CO and a dynamic range of 10 for 13 CO and C18 O. For the continuum
5
Max Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany
at 850 µm, the assumed dynamic range is 30. For all isotopologues, the62
34
PurpleMountainObservatory
andKey
LaboratoryofRadio
Astronomy,
ChineseAcademyof
Sciences,
2 out
Westout
Fig. 9. From top left to bottom right: CO ice abundance in the STN, ↵ = 10 , 10 , and 10 models. We note the di↵erent vertical scale fromratio R
CO /Rmm is a steep increasing function of disk turbulence.
othersimilarplots.
Beijing Road, 210008 Nanjing, China
2

2.1.

Millimeter Observations of the disk around GW Ori
M. Fang1,2 , A. Sicilia-Aguilar3,1, D. Wilner4 , Y. Wang5,6 , V. Roccatagliata7 , D. Fedele8 , and J.Z. Wang91

Departamento de Fsica Teórica, Universidad Autnoma de Madrid, Cantoblanco 28049, Madrid, Spain
Department of Astronomy, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
3
SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
A&A proofs: manuscript no. GW_Ori_SMA
4Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
5
Max Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany
6
Purple Mountain Observatory and Key Laboratory of Radio Astronomy,
Chineseof disk
Sciences,
2 WestM. Fang et al.:Millimeter Academy
Observations
thearound GW Ori•of 三重星の前主系列星(PMS)の共通円盤のCO輝
Beijing Road, 210008 Nanjing, China
Table 1. Coarse grids of the parameters for modeling disk continuum emission. 線とダスト連続波をSMAで観測；分解能
7
Universitäts-Sternwarte München, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679, München,
Germany8
Range
Max Planck Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85748, Garching, Germany
1arcsec ̶> 103AUスケール
2

観測

モデル

Parameters

Input
Central starEﬀective temperature (T eﬀ )a5500 K
Stellar radiusa
7.5 R⊙
12 mass (M⋆ )aStellar
3.9 M⊙
Disk
Inner radius (Rin )a
1.2 AUGap size (Rgap )a45 AU
anuscript no. GW_Ori_SMA
Dust depletion factor (ξgap )a3×10−3Characteristic scaling radius (Rc )250, 270, 290, 310, 330, 350, 370, 390 AU
uum emission.
Surface density gradient parameter (γ)
0.1, 0.3, 0.5, 0.7, 0.9, 1.1
Angularscale
height(h)
0.15,0.17,0.19,0.21,0.23
c
Range
Ψ
0.1, 0.15, 0.2
2
2
Input
(χtotal − χtotal, best < 0.5)
Best-fit
Disk
mass
[1,
1.5,
2,
2.5,
3,
3.5,
4, 4.5, 5, 5.5, 6, 6.7, 7]×10−2 M⋆
320, 330, 340, 350 AU
300–340 AU
320 AU
Disk inclination (i)30◦ , 40◦ , 50◦.2, 0.3,
0.4,The
0.5 observed position-velocity
0.1–0.4 diagram from 0.2
aFig.
6. (a)
the 12 CO J = 2 − 1 map along the north-south
direction
cross
GW
Ori.
The
oﬀset
refersPosition angle
10◦6, 0.17, 0.18, 0.19
0.15–0.19
0.18
to the distance along cut from GW Ori. The contours start at 0.15 Jy beam−1 (3σ) with an interval of 0.05 Jy beam−1 (1σ). The dotted, dashed, and1, 0.15, 0.2
0.1-0.2
0.1(a) disk inclinations of 20◦ , 40◦ , and 60◦ , respectively. (b) Samedash-dotted lines−2show a Keplerian rotation
curve with a central
mass of 3.9Notes.
M⊙ withFixed parameters.
3.3, 3.4, 3.5]×10 M⋆
3.0–3.4×10−2 M⋆
3.1×10−2 M⋆◦diagram◦◦0as◦ , in35Panel, 40◦ (a) but for the modeled30position-velocity–35◦
35◦ with a disk inclination of 35 .

(χ2total − χ2total, best < 2)

• GW Ori A/B/C：Bは分光連星(sep.~1AU), C
はsep.~8AU• ３重PMSを囲む円盤の⼲渉計観測はおそらく290–350 AU
0.1–0.5
これが初。
0.15–0.19
0.1–0.2
3–3.5×10−2 M⋆
30◦ –40◦• GW Ori A: M*~4Msun, dM/dt~3e-7 Msun/yr• 結果：SMAビームでは共通円盤を空間分解でciently constrain Mdust and Rc . For each model, the goodness of
きないので、(i)ダストの強度分布,(ii)分⼦ガWith theparameters of the above best-fit
model, we can comthe fit χ2mm is calculated bypute the model map of the 12 CO J = 2 − 1 velocity-integrated
intensities,
and compare it with the observation.
However,
to calスの速度構造(図6),(iii)
SEDをモデリング。
!
4.2. CO emission

Table 3. Disk model parameters for modeling dust and gas emission ofouter
p as disk (R > Rgap ). In Fig. 7, we show the opacity spectra for
GW
Ori.populations derived from Mie calculations.
the
two
dust
tant,

lecmo- The

Parametersstellar parameters
adopted in theValues
models are
Central starT eﬀ =5500 K, R⋆ =7.6
R
,
and
M
=
3.9
M
,
taken
a
⊙ temperature⋆⊙Eﬀective
5500 Kfrom Paa
per I. We use the RADMC–3D
code(version
0.40,
Stellar radius
7.5 RDullemond
⊙
a
2012)
transfer(9) to do the radiative
Stellarmass in the disk models,
3.9 M⊙ and vary
the free parameters to calculateDisk
the continuum emission maps
Inner radius(Rin )a
1.2 AUthe model
at 1.3 mm and the SEDs.
For simplicty,
we convolvemeterasizewith(Rgap )the
AU for theules
continuum
emission Gap
maps
synthetical45beam−3
Dust
depletion
factor(ξ)
2.5×10gap
ince continuum emission map to simulate
SMA
the observations.

Characteristic scaling radius (Rc )320 AU
disk
Surfacedensity
gradient
parameter
(γ)
0.2
e the
Angular scale height (hc )
0.18
30◦ ,
Ψ continuum emission0.1
d4.1.2.
us- Scheme for modeling
Disk
mass
0.12 M⊙
nonT
200 K
atm,100
AU
ated
a
Power
for T atm (ζ)−0.5
In order to model
theindex
continuum
emissionfromthe disk ofwithδa
GW Ori, we have 10 parameters.
Among them, the 82 ◦parameters,
withDisk inclination (i)35 structure.
Rthe
in , Rgap , ξgap , Rc , γ, hc , Ψ, and Madust , are for the disk◦
Position
angle
10
quaAs discussed above, we have fixed Rin =1.2 AU, Rgap =45 AU. To
peakcomparethe(a) model results with the observations, we need to
the Notes. Fixed parameters.
know
ob- the orientation of the disk with respect to the observer,
which
can be characterized with two more parameters, the incliion. intensity
along the east-western and south-northern direction◦

1

χ2mm = NN2i=1 (µi − ωi )σ2culate the CO emission, one needs to know the gas temperaure(5)distribution in the disk. At the first step, we assume that the gas• 表３のように諸元を推定。

and dust has equal temperatures, and ignore that CO molecules
are frozen out of the gas phase when T < 20 K and that thewhere σ is the noise of the observed 1.3 mm map, N the numbergas temperatures may be higher than thegasdust temperatures nearof pixels with the values above 3σ, µ the modeled data, and ωthe disk surface. The velocity fields of gas material in the disk
the observed data. By comparing with the observations, we ob-models are assumed to be Keplerian. The thermal line broadtain sets of disk models providing a good fit which we define asening is automatically included in the RADMC–3D code. Beχ2mm − χ2mm, best < 2 where χ2mm, best is the minimum χ2mm amongsides it, we also include the turbulent line broadening by assumall the models. In total, we have 102 good-fit models. Then, weing a constant linewidth of 0.01 km s−1 from turbulence. Takcalculate the SEDs of the good-fit disk models in the range froming an abundance ratio 12 CO/H2 = 10−4 , which is the canonabundance of interstellar medium (ISM), we calculate the70µm
1.3 mm,and(κabs
comparethem andwithκscatheof GWforOri.ical
Fig.
7. to
Opacityspectrafor absorption
foronescattering)
122COJ = 2 − 1 line emission using the RADMC–3D code withthe small
dust populationline)
and thelarge
population
The
goodnessof the fit(solidof theSED
χSED
is dustcalculatedby(dashan assumption of local thermal equilibrium (LTE) conditions.
line) used in our disk modeling.
The resulting maps are simply convolved with the correspondingNsynthetical beams, and then compared with the observation. We2"1(µi − ωi )find that the predicted peak intensity of 12 CO J = 2 −1 is about 3
χ2SED =
(6)2N
σtime weaker than the observation. We vary the parameters among
with χ2total −i=1
χ2total, besti < 0.5. The disk model with minimum χ2total the good-fit models listed in Table 2, and the model emission foris considered
the best-fit
where
N is theasnumberof model.
the observed wavelengths, µ the syn-12 CO J = 2 − 1 line are all 2–3 time weaker than the obserthetic flux density, ω the observed flux density, and σ the obser-vation. The previous studies have shown that the gas tempera-• opacityスペクトル(図7；左)は、No.11の論⽂と⽐較できそう。No.11の論⽂と⽐較するた
めには２次のモーメントマップ必要(プレゼンなし)• 共通円盤内におそらく存在するであろう、個別の円盤の検出、空間分解はALMAに期待。

Age spreads and the temperature dependence of age estimates in Upper Sco
Qiliang Fang1,2 , Gregory Herczeg1 and Aaron Rizzuto3

3.1.

1

Kavli Institute for Astronomy and Astrophysics, Peking University, 100871 Beijing, China
Department of Astronomy, School of Physics, Peking University, 100871 Beijing, China
3
Department of Astronomy, University of Texas, Texas, USA2

6E-mail contact: gherczeg1 at gmail.com

Fang et al.

Past estimates for the age of the Upper Sco
6 Association are typically 11–13 Myr for intermediate-mass
Fang et al. stars and 4–5
Myr for low-mass stars. In this study, we simulate populations of young stars to investigate whether this apparent
dependence of estimated age on spectral type may be explained by the star formation history of the association.
Solar and intermediate mass stars begin their pre-main sequence evolution on the Hayashi track, with fully convective
interiors and cool photospheres. Intermediate mass stars quickly heat up and transition onto the radiative Henyey
track. As a consequence, for clusters in which star formation occurs on a similar timescale as the transition from a
convective to a radiative interior, discrepancies in ages will arise when ages are calculated as a function of temperature
instead of mass. Simple simulations of a cluster with constant star formation over several Myr may explain about halfof the diﬀerence in inferred ages versus photospheric temperature; speculative constructions that consist of a constantstar formation followed by a large supernova-driven burst could fully explain the diﬀerences, including those between
F and G stars where evolutionary tracks may be more accurate. The age spreads of low-mass stars predicted fromthese prescriptions for star formation are consistent with the observed luminosity spread of Upper Sco. The conclusionthat a lengthy star formation history will yield a temperature dependence in ages is expected from the basic physics ofpre-main sequence evolution and is qualitatively robust to the large uncertainties in pre-main sequence evolutionarymodels.
Fig. 6.— Emperical isochrone of Upper Sco
members
comparedwith oftheUppermodel
from thewith
Dotter,the magnetic
Fig.
6.— Empericalisochrone
Scoisochrones
members compared
the model
isochronesFeiden,from theand
Dotter, the magnetic Feiden, andSiess pre-main sequence evolutionary models, with isochrones of 1, 3, 10, and 15 Myr. Compared to the isochrone of Pecaut and Mamajek

Siess pre-main sequence evolutionary models, with isochrones of 1, 3, 10, and 15 Myr. Compared to the isochrone of Pecaut and Mamajek
Acceptedby
ApJ(2016)
in is
theirsimilarthe isochrone
hereKis andsimilarfor starsfor
hotterthan
4000 K and is fainter for cooler stars.
(2016) in their similar Figure 5, the isochrone
here
similarforFigurestars 5,hotterthan 4000
is faintercoolerstars.

http://arxiv.org/pdf/1705.08612 additions to the Smooth-Quench model could become
additions to the Smooth-Quench model could become【背景と動機】

hopelessly
complicated,hopelessly complicated, adding a burst
at theend of adding a burst at the end of
the Smooth-Quench
model
maximizes possible
agethedif- orbital evolution
Tidal
dissipationin rotating
low-massstars
and implicationsfor
Stellar
Association(群落)の年齢はHRで決めるが、低質量星と等時線
the• Smooth-Quench
model
maximizes
possibleage
differences
betweenstars
of di↵erent
temperatures,
therebyofclose-inplanets.
From temperatures,
thePMS
totherebytheRGB
at solarmetallicity
ferences
betweenstars ofI.di↵erentの⽐較から求めた年齢と⼤質量星の転回点から求めた年齢が必ずしも⼀
probing the limits of age di↵erences versus spectral type.
1 limits of age2,3
2
1,4
1,5
probingtheversusspectraltype.
F.
Gallet, E. Bolmont di↵erences, S. Mathis, C.
Charbonnel
Amard
The
Smooth-BurstmodelandalsoL.
may
represent a reason致しない。これは本研究の対象のUpper
Sco以外でも確認されている。
a reasonscenario
of a pasta burst in
1The Smooth-Burst model also may represent
Department of Astronomy, University ableof Geneva,
Chemindessupernova
Maillettestriggering
51, 1290 Versoix,
Switzerlandstarformation
(see
§4).
The
acceleratingstar
formation
scenarioofapast
supernova
triggering
a
burst
in
2able
• 差の要因：⾒落としている連星の寄与、そもそも群落のメンバー星が適Laboratoire
AIM Paris-Saclay, CEA / model
DRF - of
CNRS -andUniv.
Paris(2000)
Diderot- IRFU/ SAp, Centrede Saclay, F-91191
Stahler
is notinvestigatedinstar formation (see §4). The accelerating
starPalla
formation
but wouldinyield results that are intermediate
切に同定されていない可能性も⾼い。
modelof Palla and Stahler (2000) is thisnot work
investigatedbetweenthe Smooth-Quench and Smooth-Burst models.
this work but would yield results thatare intermediate
13 summarized in Figure 7. The
These models are
between the Smooth-Quench and Smooth-Burstmodels.
【解析】
Smooth-Quench model is described by two parameThese models are summarized inters:
Figure7. sinceThe
the time
the association first started forming• GAIAデータを使⽤,
NIR⾊超過から円盤を持つ天体を排除しSmooth-Quench
model is 減光補正,
describedstars,
byparamettwo
,
and
the
time since the star formation wasstartters: たのち、M*=0.1
the time since the association
firststartedformingquenched,tquench. The decrease in star formation after
to 2.0 Msunの若い星1e4個をメンバーとして同定.
stars, tstart , and the time since the quenchingstar formation
was by e t2 . The star formation rateis described
Fig. 7.— An illustration of the star formation histories adopted
quenched,tquench . The decrease in starformationafter• Teﬀのビン内の星々の値を平均して図6を作成；等時線と⽐較；図6の⾚is thenanalyticallydescribed byin this work. In this example, the age of the start of star formation2in the association, tstart , is 15 Myr and the time since the start of(quenchingis described by e t . The star formation
rate
い曲線の意味の説明が⾒当たらないのだが、これが後の議論の鍵?
the burst/quench(tquench )adopted
is 5 Myr.
Fig. 7.—of the starformation histories
1t Ant illustrationt
startquenchAge spreads and the temperature dependence of age estimates in Upper Sco
Qiliang Fang1,2 , Gregory Herczeg1 and Aaron Rizzuto3

Age discrepancies

3.2.

1

Kavli Institute for Astronomy and Astrophysics, Peking University, 100871 Beijing, China
Department of Astronomy, School of Physics, Peking University, 100871 Beijing, China
3
Department of Astronomy, University of Texas, Texas, USA
2E-mail contact: gherczeg1 at gmail.com
Past estimates for the age of the Upper Sco Association are typically 11–13 Myr for intermediate-mass stars and 4–5【解析(続き)】Myr for low-mass stars. In this study, we simulate populations of young stars to investigate whether this apparent• ChabrierのIMFを与えて、星のpopulation合成；その際、図７のような
dependenceof estimated age on spectral type may be explained by the star formation history of the association.
Solar星形成史(しずしずと星々を⽣んでいるが、突如爆発的星形成したの
and intermediate mass stars begin their pre-main sequence evolution on the Hayashi track, with fully convective
interiors and cool photospheres. Intermediate mass stars quickly heat up and transition onto the radiative Henyey
ち、⼀気に星形成活動停⽌)を仮定。
track. As a consequence, for clusters in which star formation occurs on a similar timescale as the transition from a
convective
to a radiative interior, discrepancies in ages will arise when ages are calculated as a function of temperature• 爆発的星形成を特徴付けるパラメタA（って何？）を振って、群落中の
instead of mass. Simple simulations of a cluster with constant star formation over several Myr may explain about half星の年齢の有効温度に対する依存性を調べた（図８）。
of the diﬀerence in inferred ages versus photospheric temperature; speculative constructions that consist of a constantstar
followed by a large supernova-driven burst could fully explain the diﬀerences, including those between• formation
図６(観測)と図8(理論)の⽐較から、星形成史と年齢のばらつきを議F and G stars where evolutionary tracks may be more accurate. The age spreads of low-mass stars predicted fromthese論。
prescriptions for star formation are consistent with the observed luminosity spread of Upper Sco. The conclusionthat a lengthy star formation history will yield a temperature dependence in ages is expected from the basic physics ofpre-main sequence evolution and is qualitatively robust to the large uncertainties in pre-main sequence evolutionarymodels.
【疑問＆感想】
Accepted by ApJ• 観測されているHR図は、Smooth SF̶>Starburst̶>Quechingで確かに説明できそう；しかし、なぜstarburstでなければいけないのが納得できなかった。例えばaccerlating(Palla& Stahler)などではダTidal
dissipation in rotating low-massSFstars
andimplications for the orbital evolutionof close-inplanets. I. From the PMS to the RGB at solar metallicity
メ？
http://arxiv.org/pdf/1705.08612

F. Gallet1 , E. Bolmont2,3 , S. Mathis2 , C. Charbonnel1,4 and L. Amard1,5
1• 図6のHR図はビニングし過ぎ？(e.g., ヘニェイトラックすら⾒えない).

Department of Astronomy, University of Geneva, Chemin des Maillettes 51, 1290 Versoix, Switzerland
2 • イントロで述べている⼀般的な問題認識と本研究の「結論」がリンクしLaboratoire AIM Paris-Saclay, CEA / DRF - CNRS - Univ. Paris Diderot - IRFU / SAp, Centre de Saclay, F-91191ていない(そもそもhigh-massはサンプルに⼊っていない)。
「Discussion&Conclusions」はたいへん読みにくい。私がレフェリー
13
だったら、以上２点は修正要求するかな。

Fig. 8.— Age versus temperature for the median star in atemperature bin in our simulations. The top panel shows thatage di↵erences are large for a model with tstart = 16 Myr andtquench = 4 Myr, while these di↵erences are reduced for a modelwith tstart = 16 Myr and tquench = 8 Myr, i.e., less of an age
spread and star formation quenching further in the past. In eachpanel, simulations with di↵erent A values are labeled by di↵erent
colors and line styles.

Tidal dissipation in rotating low-mass stars and implications for the orbital evolutionof close-in planets. I. From the PMS to the RGB at solar metallicity
F. Gallet1 , E. Bolmont2,3 , S. Mathis2 , C. Charbonnel1,4 and L. Amard1,5
1
2

4.1.

Department of Astronomy, University of Geneva, Chemin des Maillettes 51, 1290 Versoix, Switzerland
Laboratoire AIM Paris-Saclay, CEA / DRF - CNRS - Univ. Paris Diderot - IRFU / SAp, Centre de Saclay, F-91191

【動機】

13

• 星と惑星の相互作⽤；重要性は認識されているが、星の進化モデルにおける惑星の影響(⾓運動量の時間発展)を評価す
るモデル群(2012-15年に多数)でさえも組み込まれてこなかった。
• 潮汐散逸モデル(Mathis 2015)と星の進化モデルの計算コードSTAREVOLが近年⼤きく改善；両者と統合したい。

F. Gallet et al.: Tidal dissipation in rotating low-mass stars and i

mo
lag
Bo

【予備知識】• ⼩質量星の恒星⼤気の対流envelopeの進化に惑星は影響を及ぼす；主要メカニズムは２つ。

tia
are
tid
is
(20
qu
tem
fre
the
cie
Ω⋆
fec
rad
wa
Ter
cel

1.“equilibrium tide”(平衡潮汐？)：恒星⼤気の⼤局的な進化に影響；1960年代に指摘されエネルギー散逸率は⼀定として扱われてきた。
2.“dynamic tide”(動的潮汐？)：星内部へ下降(降着)する
対流による摩擦、コリオリ⼒起源【計算】• 以上の効果をSTAREVOLコードに実装；PMS ̶> ZAMS
̶> Teminal Age MS (TAMS)まで、M*=0.3 ̶ 1,4 Msunの星について計算。

TAMSと呼んでいるけど、この分野ではRGBもMS扱いなの？??

Fig. 1. Stellar evolution tracks in the Hertzsprung-Russell diagram forthe rotating models of 0.3, 0.4, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and
1.4 M⊙ at solar metallicity. We show the evolution up to the RGB for
the more massive stars or up to the evolution stage the models reach at
20 Gyr for the less massive stars. The symbols represent the first step ineach evolution sequence (triangle), the ZAMS (square), and the TAMS
(cross).

TAMSはcrossでなくて三⾓形?

in
tio
riu
tio
on
rat

4.2.

Tidal dissipation in rotating low-mass stars and implications for the orbital evolutionof close-in planets. I. From the PMS to the RGB at solar metallicity
F. Gallet1 , E. Bolmont2,3 , S. Mathis2 , C. Charbonnel1,4 and L. Amard1,5
1
2

Department of Astronomy, University of Geneva, Chemin des Maillettes 51, 1290 Versoix, Switzerland
Laboratoire AIM Paris-Saclay, CEA / DRF - CNRS - Univ. Paris Diderot - IRFU / SAp, Centre de Saclay, F-91191

【結果】

13

• 潮汐散逸を考慮した星の進化

A&A proofs: manuscript no. aanda

ZAMS「星」の半径:太陽半径単位「星」の回転各速度:
太陽のそれに対する⽐

ZAMS

TAMS

TAMS

A&A proofs: manuscr

Fig. 2. Top-left: Evolution of the stellar radius R⋆ of stars from 0.3 to 1.4 M⊙ as a function of time. Top-right: Surface angular velocity (scaled tothe present Sun angular velocity Ω⊙ = 2.87 × 10−6 s−1 ) evolution for the diﬀerent stellar masses. Bottom-left: Evolution of the radius aspect ratio
α = Rc /R⋆ of stars from 0.3 to 1.4 M⊙ as a function of time. Bottom-right: Evolution of the mass aspect ratio β = Mc /M⋆ of stars from 0.3 and1.4 M⊙ as a function of time. The symbols represent: the first step in each model (triangle), the ZAMS (square), and the TAMS (cross). Table 1
summarize at which evolutionary step the models of this work stop.

【潮汐散逸はどの進化段階でもっとも効くか？】
orders of magnitude during the evolution of stars. This may•eral潮汐散逸を特徴づける量「星」の年齢1M のZAMSof the time-dependent tidal potential proportional to the spheri- suncal harmonic Ylm ) that gives the ratio between the perturbation oflead to weaker dissipation rates.
A&A proofs: manuscript no. aanda
In the case of a coplanar star-planet system in which the orbit the gravitational potential induced by the presence of the planat the stellarof the planet is circular, the frequency-averaged tidal dissipation etary companion and the tidal potential" evaluated#2(Ogilvie 2013; Mathis 2015) is given by:
surface. Its imaginary component Im k2 (ω) is a direct estima! +∞ "tion of the tidal dissipation. The interest of this formalism is that# dω 100π $ α5 %222
(1 − γ) (1) it is possible to decompose Eq. 1 into two parts: the factor ϵ 2 on
Im k2 (ω)< D >ω =
=
ϵ
sun
5
ω
63
1−α
−∞
the one hand, and the part of Eq. 1 that is a unique function of
$
%2 &
$
% '
α and β on the other hand. The first part takes into account the
3 3
1−γ 3
3
4
× (1 − α) 1 + 2α + 3α + α
1+αrotation rate of the star (via ϵ) and the second part only takes
2γ
intoaccount the dependence on the internal stellar structure (via
&
$
%
'−2
the structural parameters α and β). As in Mathis (2015), we can
3
5
1
3
9
× 1+ γ+
1 + γ − γ2 α3 − (1 − γ) α5 ,therefore
express the frequency-averaged dissipation at a fixed
2
2γ
2
2
4
rotation:• 添え字”c”は星の放射核with#"• 添え字”e”は星の対流外層RcMc
ρe α3 (1 − β)
< D >Ω= ϵ −2 < D >ω = ϵ −2 < Im k22 (ω) >ω ,
(3)
ω
< 1.
α=
, β=
, γ = Fig. =2. Top-left:3 Evolutionof the stellar(2)radius R⋆ of stars from 0.3 to 1.4 M⊙ as a function of time. Top-right: Surface angular velocity (scaled toR⋆M⋆ρc β(1 − α )the present Sun angular velocity Ω⊙ = 2.87 × 10−6 s−1 ) evolution for the diﬀerent stellar masses. Bottom-left: Evolution of the radius aspect ratio
α =of
Rc /R
from 0.3 to 1.4 toM⊙ theas a functionof time.
Bottom-right:of theratio
β >=ω (left)
Mc /Mof stars modified
from 0.3
andFig.Evolution
6. Frequency-averaged
tidalaspectdissipation< D̂and⋆ equivalent
tidal
quality factor Q′ = 3/(2 < D̂ >ω ) (right) as a function of⋆ of stars
that only
depends
on α andβ.
We canalsomass
definea second
k22 is the Love number
degree2 correspondingin linearscale. The thesymbolrepresent:
the ZAMS
and the (cross).
TAMS (cross).
M⊙l as
The symbols represent:
the first stepdissipationin eachtime
model(triangle),
ZAMS(square),and(square)
the TAMS
Table 1
frequency-averaged
using
the criticalangularquadrupolar mode (klm ,1.4
with= a2 function
and m =of2time.
the components
Fig. 6. velocFrequency-averaged tidal dissipation < D̂ >ω (left) and equivalent mosummarize at which evolutionary step the models of this work stop.

1M の
TAMS

5.1.

NumberNumber

Lacc (L⊙ )M⋆ (M⊙ )dM⋆ /dt(10−6M⊙ /yr)kthirley et al. 2002) or by fitting
γ = 6.4 × 10 γ M⊙ yr
Ṁ0p =/G,(21) (8)≡ p̄(t)α(x) ,= k t α(x)static outer envelopes (B = 0 in eq. (13)) and central fre
4πG
t priorassumptionsof temper62014).
The
last
one
is
the
dynamic
asymptotic
free-fall
soVariable
protostellar
mass accretion
rates
in
cloud
cores
∼
mean
molecular
weight
is
µ
2
for
H
fall
collapsingcores (14),with Ratedimensionlesscentral ma=
γ=1.2
2 MCs and
Mass Accretion+. 2005; Hung1 et al. 2010). Vari-2,3 where the
3/2 3n−2lutiontowards the MC core centre as x → 0 :
k
t
γ=1.1
Yang Gao and Yu-Qing Lou
M⊙
solar!mass.
The≡
relevant
dynamic
time-scale
1.2 and 1.
Mis=them(x)M̄ (t)m(x)"1/2!,"1/2 td = (9) m0 = 0.975, 41.25, 1.15 and 0.26 for γ = 1.0, 1.1,
ve been adopted in fitting theseγ=1.05(3n − 2)G
12mEngineering,(3nUniversity,− 2)m0 Beijing 100084,respectively
0.6 × 10 yr is estimatedby density
scale
(17).
(Gao, Lou & Wu 2009). Time evolutionof pr
0
Centrefor Combustion
EnergyTsinghua
China
γ=1.3
bert
spheres(Alves et al.
2001;and Departmentvof=Thermal
−
,
α=
,
(14)
3
2
2
(3n − 2)x
2x
Department of Physics and Tsinghua Centre
Astrophysics(THCA),
Tsinghua
University,
Beijing
China MAR Ṁ⋆ , protostellar mass M⋆ and accretion lwith µforandmH being
the meanmolecular
weightand100084,
the tostellarlaw
3 models (van der Tak et al.
Tsinghua-NAOC joint Research Center for Astrophysics,
Tsinghua
University,
Beijing
100084,China
whereintegration
constant
mγ0 is̸=the
reducedenclosedhydrogen
mass,
n
>
2/3
and
4/3.
For
the
CPmass
EoS, minosity Lacc0 are shown in Fig. 1.
Mueller et al. 2002; Hung et al.
+
Protostellar
Mass panel in Fig. 1 and MA
A solid0.6conclusion of
the topthedimensionless
3 weDECREASING
RATE mass
E-mail contact:
mail.tsinghua.edu.cn
haveas nx →+ 0γ ,=representing
2MASS
and n ACCRETION= γ= 1 is anprotostellar
isothermalgas.
2013),
doublegaoyang-00
power atlaws
or MAR [see eq. (15)]. Invoking central free-fall solution (14)(22) is that0.4CP protostellar MARs decrease with increa
5 are revisited
Sphericalhydrodynamic
modelspolytropic
equation
of
state
(EoS)
for
forming
protostars
inorder
The dimensionlessindependent
self-similarvariablexisas (Hogerheijde& Sandell
2000;with aReferring【総評】
to MAR
scaling
tdwe=derive
0.6 ×the10central
yr, theandrecalling
eqns (9)(21)
and at(10),
protoing
time t for γ > 1, and in contrast to constant MARs
to investigate
the so-called
luminosity protostellar
conundrum
highlighted
byxanyobservations.
a molecular
cloud
(MC)
+
combination
of0+r forandt, 0andα(x),aredi-core with
0.2
t al.
2003; Kurono
et al. 2013).
stellar
MAR
as
→
:
(r →
time
t > For0, m(x)i.e.,
x and→ 0+v(x)) MAR
isothermal
EWCSs (γ = 1 and n = 1), oﬀer a sensible resolu
such
an
EoS
with
polytropic
index
γ
>
1,
the
centralmass
accretion
rate
(MAR)
decreaseswithincreasingtime
as• 分⼦雲コアの構造と進化を説明するポリトロープ球モデルの論⽂群のひ
mensionlessreducedmass
density,enclosedmass
andradialic self-similar
polytropic sphere¯
3/2 3(n−1)(15)
can beestimated
by
Ṁ
(t)
=
ϵ
Ṁ
(t),
viz.
⋆ ≡ Ṁ (t)m00tion to the luminosity
problem.
The evolution of protostellaṀ0this= k luminositytm0problem.
/G(15)protostellar
0 As the0a protostar emerges, oﬀering a sensible solution to
MAR decreases, the10
Accretion Luminosity
velocity,
respectively.
By
massconservation(1),
wederiveowluminosity
theirとつらしい。この研究は何がスゴいのか、まったく理解できなかった。
mass
densityandtem!
"masses (middle panel in Fig. 1) shows that most protost
also decreases, meaning that it is invalid
to &infer the3(1−γ)
star formation time
the currently observed(see Lou
2008,fromfor details).
t Gao 2006; Wang & Lou−6−1
comparison
withpreviousanal2
massesin their early phase. For γ = 1.
luminosity
using an isothermal model.ṀFurthermore,
observationsof radialprofilesradio
continuaof are accumulated(t) ==ϵ α(x)x× 10 mM⊙andyr the. scaling(22)
この分野の重要な論⽂が適切に引⽤されておらず、本研究の位置づけが⋆0dimensionalm(x)[nx− v(x)].6.4 density(10)
To
estimatethetime-dependent
5
5 yr
0.6
×
10
numerous MC cores evolving towards protostars alsosuggestthat polytropic
dynamic(11)spheresof γ >
should beused ∼ 80% of the stellar mass is accumulated in the fir
nearlyfactorsof variables
in eqns. (5)−(9),and (15),
we1need
不明。
Thethegasempiricaltemperature
T fromgas
law is to Myersin physical models.
∼ 104 yrs of0 formation. By protostellar accretion lumino
informationforϵtheMC
cores.
AccordingHereeﬃciency
coeﬃcient∼ ideal0.3, assuggested
by
1
1.5
2
Evans
et al. (2009),
a typical
radiusof thata starity (24), only0for γ >0.5
7/6 dynamicµm(2005)
p andµm
Acceptedby MNRAS
H(2007),H
2n−2the fraction
γ−1
γ−1
Alves
describes
of(t)α(x)materialst (105 ) yr CP spheres do accretio【モデル構築】3
4
T =et al.
=
kt
α(x)
≡
T̄
,
(11)
MC core is ∼ 0.01 − 0.1 pc (∼ 10 − 10 AU); andluminosity decrease with increasing time (bottom panel
kforming
kB
ends up onto(see also Evans et al. 2009). By t
B ρ a protostar
http://arxiv.org/pdf/1705.09012
ROPIC
GAS SPHERESa typical mean number density in MC cores is estimatedFigureTime evolution showof protostellar
MARs dM∗ /dt,
massesFig.
1). 1.Observations
that bolometricluminosity• 状態⽅程式；妥当？ integrationbyof∼eq.
protostellar
mass
M
(t)
becomes4(22), 5the −3⋆Mandaccretion
luminosityLofdynamic
CP
MC
cores
of
10 cm . Notingthat thepartsof t∗
acc
where kB is 10
the−Boltzmann
constant.
Alldimensionlessbarred factors
star-forming
MC
cores
do
roughly
decrease
in
their
ties
of
star-forming
MCsandIdentifying
andAnalysing Protostellar
Disc
Fragmentsin
Smoothed
Hydrodyvarious γ for protostar formation. For γ > 1 CP MC cores, MARs ear!radius"3(1−γ)of bothandnumberdensity
are aroundParticle
unity in relaGao
and
Y.-Q.
Lou
are of therespective
physicaldimensions.
phases0, I) of timestar (top),formation(fig. 13accumulate
in Evans et aϵt (5) and (7),t we may choose the length
decrease(Class
with increasing
and protostars
namicsthe
Simulations−6scale and number
rvations,
general polytropictions
M⋆ (t) =
6.4
×
10
m
M
.
(23)
0
⊙
There
are
three
analytic
andasymptotic
solutionsfor•輻射輸送も解かない.
most ofsuggesting
masses in theirof formation(middle). In2009),
γ >early
7/6 phases
dynamic
CP collapses.
3,4
4density− 3γ1,5scale
0.6 ×
as105 yrformation
model
(Wang
&
Lou
CassandraHall, Y.
DuncanForgan,Ken
Rice
4and1,2
Gao
and
Y.-Q.
Lou
function
(PLF)
PMF,
we
derive
model
PLF
the
bottompanel,
onlyforlargervaluesofγ=
1.2
and 1.3 do
CP spheres 1/2
that
are importantwhen comparing our hydroWith
dynamicCP
accretion
model
in
hand,
we cou
n
3
40
&
Lou
2009;
Lou
&
Hu
2010,
1
3
accretion luminosity decrease with increasing time.
r̄ detailed=
k t with=
4 ×observations.
10 transfer
AU(16)
e withthe observed
PLF.
By sampling
thedySUPA,
Institute for
Astronomy,
Universityof Edinburgh,Blackford
Hill,, Edinburgh,
EH9
Bypassingenergyprocesses,
we3HJ,
dynamic
model
The first
isestimate
theUK
singularObs.
evolve either from core mass function (CMF) for MC cor
2 withoutheremagnetic
field.
Hy•results
形成途上の星の半径
model
of of
Fig.
1, onefit PLF
ofpro2 −1
4 by−3 UKγ=1.2
Department
Physics&may
Astronomy,
UniversityofLeicester,
Leicester,
LE1
7RH,
the
protostellar
accretion
luminosity
35
N̄ =sphere(4πGµm(SPS)= 9 × 10 cm ,(17)
polytropic
with
Ht )
to protostellar mass function (PMF) or from initial ma
luminosity
functionh envelopes(Evanset al.of2009;
Kryukova(PLF)et al. and PMF, we derive model PLF
ialequations(PDEs)
spheriγ=1.25!
"6(1−γ)% 30&−1/n
40 2 (IMF)
functionfor stars
back to PMF; we follow the la
も定数扱い.
respectively.
Scalesofsamplingotherphysicalvariables
can be exGM⋆ (t)Ṁ⋆ (t)γ=1.3overall energy
conservation,
the withaccretion
power
2
to compare
theobserved
PLF.
By
the
dyt
ϵ
t
Obs.
Lacc (t) = pressed in14the following nmanner accordingly:×
10−4between
m20 L⊙ , protostella(24)
ter= path.
By0.6
setting
up the 4.2
relationFitx−2/n ,5 yr
ossly converted totheobservedbolometric
lumiv
=
0
,
α
=
R
⋆
(4
−
3γ)
×
10
25
namic
CP
model
results
of
Fig.
1,
one
may
fit
PLF
of
proγ=1.2
35− n)(3n − 2)h Lbol ∂M∼ L+. ∂M
As onecannotfollow the lumi- ū = k1/2 tn−1 =2(2accu
0.30 km s−1 ,
(18)
=
0
,
(1)
20
tostarswithenvelopes
et %al.000,
2009;
Kryukova&−1/net al.
where L⊙ = 3.9 × 1033 erg s−1 isγ=1.25
the solar luminosity,c analy⃝
0000(Evans
RAS, MNRAS
000–000∂ta single∂rprotostar,ution ofa statistical2
3/2 3n−2n30(3n−2)/nradius R⋆ is assumed
0.41
k t
γ=1.3 at ∼ 3R⊙ withge
samplewould By
hint overall
at evolution
track conservation,2012).
energypowerx, (19)(12) and the protostellar
15=the accretion• 1結果は図１のようになるが、これは昔からわかっていたことでは？
M⊙ ,
M̄m
== n
u enough∂p
GM
10R⊙ = 6.9 × 10 cm being the solar− 2)(3n − 2(22)G − n)(3n(3n − 2)= − formation.
−may,We grossly(2) aconverted
protostarassumecontinuous to the
Fitradius (Evans et al.
beobservedbolometriclumi25∂rstart rateρ ∂rof starr 2formation.
102009).
Realistically,
R
may
vary
duringa mass accretion⋆
Then the IMF of T̄ = µm kt2n−2 /k = 21 K ,• このモデルによると、図２のように観測を説明できるそうだ。単⼀星を(20)
HB
a globallystatic equilibrium
nosity (Oﬀner
with L&bol∼ whichL
one cannot
follow thesolution
lumi- for a CP MC process and 3R⊙ here is just a first cut. For simplicity,
acc .isAs
brier 2005) is convolved
McKee
2011;
5
+
we 20
consider four diﬀerent dynamic CP spheres with near¯ =−1
⽣むようなガスの降着相の進化を扱う本研究では、定常で等⽅的降着を
as x /G→. The
secondnosity evolutionof coreafitssingle
a =0statistical
k3/2 t3(n−1)
6.4
× 10−6M⊙analyyrone, is the asymp(21)1). Separate
comparisonsof ourofyetṀsingular
0 protostar,(3)modelstatic outer envelopes (B = 0 in eq. (13)) and central free0 for −1
totic
solutionx → +∞ with0
1
5 and 1.2考えているようだ。それなのに図２によって、「光度問題」解決へのヒ
CP spheres
the c2denoughresults (fig.
14dynamic∼
sis ofwith
a largesample
would
hint
at
10 evolution10
where
the meanmolecular
weight is µ10track
2
for
H
MCs
and
fall 15collapsing cores (14), with dimensionless central mass=
2
Bolometric
Luminosity L⊙
al. 2009) are shown in Fig. 2. The intermediate M−2/n−3/n dynamic
is
the
solar
mass.
The
relevant
time-scalet=m=
0.975, 1.25, 1.15 and 0.26 for γ = 1.0, 1.1, 1.2 and 1.3,
d0
ulk flowントが得られるのか、理解できなかった。
velocitygasof u,a thermal
single protostar
formation.
We− assumeα=
Ax ⊙ − 3(11/n)ABx a continuous,
luminosity spectrum is better fitted by our CP 0.6 × 105 yr is estimated by density scale (17).
respectively(Gao, Lou & Wu 2009). Time evolution of pro10s M withinr steady
at higher
time t;start
G lower(n−1)/nandwhile
rate
formation.
Then
the
IMF
of
phere
of γ = radius
1.25,
and
vlumi=ofBxstar
Figure 2. PLF or number of protostars with
tostellar MAR Ṁ⋆ , protostellar mass M⋆ and accretion lu%& envelopes in lumi.s PDEs
(1) areequivalent
mass
may involve
small
contributions
from
γ
=
1.2
1−n
stars (Chabrier 2005) is 2(2convolved
2011;
range 0.1
L⊙ McKee− 20 (1L⊙−inn)a logarithmic scale: separate
minosity
5 Lacc are shown in Fig. 1.
−nosityn)A bin(OﬀnernA&2(n−2)/n(2)spheres,is the radial
momen3 PDE
dynamic
respectively.
The shift of peak
+
−
+
B
x
,
(13)
model
fits
(dash-dot,dashedanddottedcurves
for
γ
=
1.3,
1.25
A solid conclusion of the top panel in Fig. 1 and MAR
3 DECREASINGMASS
ACCRETION
Myers
2011). Separate
comparisons
of(3n
our
fits of RATE
n 1.2 CP dynamic−core2)modelﬀerentγ modelsis due toofrespectiveandmodels)nto c2d PLF observations (solid3) is theconservation
specific model mean
(22) 0is that CP protostellar MARs decrease with increas5
0
1
γ
=
1.3,
1.25
and
1.2
CP
spheres
with
the
c2d
results(fig.
14
ReferringtoMARscaling
(21)
at
t
=
0.6
×
10
yr,
the
histogram,
fig.
14
in
Evans
et
al.
2009).
The
light
solid
curve
is
a time t 10duminosity,
relatedtodiﬀerent
accretion
historiesing
for−1γ > 1, and in contrast
r the GP EoS in the form of
10 to constant MARs
10 ofwhere A
and
B
are
two
&
Shi
+ integration constants (Lou+protostellar
(rfit
→with
02. foranytimet > 0, havingi.e., x →) MAR
combined65%
ofintermediate
the sources
CP0 EoSof γ = 1.25,
nt γ as shown inof
Fig.
1. Thisetfittingthat shown
isothermal EWCSs Bolometric(γ = 1 and n Luminosity= 1), oﬀer a sensible
resoluL⊙Evans
al. suggests2009) arein Fig.
The

and 10% byof Ṁ
sources
CP viz.
EoS of γ = 1.3 and 1.2,
can 25%be estimatedϵṀ0 (t),⋆ (t) =withsphere is prominent for the early stage of PMF (15) andtion to the luminosity problem. The evolution of protostellar
part of the luminosity spectrum
isbetterfittedbyourCP
respectively.
Thebolometricluminosity
is
taken
as
the
accretion!
"masses (middle panel in Fig. 1) shows that most protostar
3(1−γ)yr), with properly weighted contributions3 fromt a termination
luminosity
withlimitlumiL.
tion,
submillimeter
continuumandThereexistsalso
aLarson-Penston
type
asymptotic−6
ter = 0.08L−1⊙solutions
dynamic
sphere
of
γ
=
1.25,
while
higherandlower
masses are accumulated in their early phase. For γ = 1.3,
of 1.2 < γ < 1.3. For illustration, a quick trial Ṁ⋆ (t) = ϵ
6.4 × 10 m0 M⊙ yr . (22)