【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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, Chinese Academy of Sciences, 2 West
Beijing Road, 210008 Nanjing, China
7
Universitäts-Sternwarte München, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679, München, Germany
8
Max Planck Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85748, Garching, Germany
212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

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
13

A&A proofs: manuscript no. aanda

4.2.

2014). The last one is the dynamic asymptotic free-fall solution towards the MC core centre as x → 0+ :
v=−

!

2m0
(3n − 2)x

"1/2

,

α=

!

(3n − 2)m0
2x3

"1/2

,

(14)

where integration constant m0 is the reduced enclosed mass
as x → 0+ , representing the dimensionless protostellar massor MAR [see eq. (15)]. Invoking central free-fall solution (14)and recalling eqns (9) and (10), we derive the central protostellar MAR as x → 0+ :¯ (t)mṀ = k3/2 t3(n−1) m /G ≡ Ṁ
(15)
0

0

0

0

(see Lou & Gao 2006; Wang & Lou 2008, for details).
To estimate the time-dependent dimensional scaling
factors of variables in eqns. (5)−(9), (11) and (15), we needempirical information for MC cores. According to Myers(2005) and Evans et al. (2009), a typical radius of a starforming MC core is ∼ 0.01 − 0.1 pc (∼ 103 − 104 AU); anda typical mean number density in MC cores is estimatedby ∼ 104 − 105 cm−3 . Noting that the dimensionless partsof both radius and number density are around unity in relations (5) and (7), we may choose the length scale and number
density scale asr̄ = k1/2 tn = 4 × 103 AU ,(16)N̄ = (4πGµmH t2 )−1 = 9 × 104 cm−3 ,(17)respectively. Scales of other physical variables can be expressed in the following manner accordingly:ū = k1/2 tn−1 = 0.30 km s−1 ,

(18)

0.41
k3/2 t3n−2
=
M⊙ ,
M̄ =
(3n − 2)G
(3n − 2)(19)

T̄ = µmH kt2n−2 /kB = 21 K ,(20)¯ = k3/2 t3(n−1) /G = 6.4 × 10−6 M yr−1 ,
Ṁ
0
⊙

(21)

where the mean molecular weight is µ ∼= 2 for H2 MCs andM⊙ is the solar mass. The relevant dynamic time-scale td =0.6 × 105 yr is estimated by density scale (17).

3

DECREASING MASS ACCRETION RATE

Referring to MAR scaling (21) at td = 0.6 × 105 yr, theprotostellar (r → 0+ for any time t > 0, i.e., x → 0+ ) MAR(15) can be estimated by Ṁ⋆ (t) = ϵṀ0 (t), viz.

!

Ṁ⋆ (t) = ϵt"3(1−γ)

6.4 × 10−6 m0 M⊙ yr−1 . (22)6420