星形成ゼミ 2015/ 2/15 Newsletter #255の26–30本目!

!

26. “The dynamical evolution of molecular clouds near the Galactic Centre – I. Orbital
structure and evolutionary timeline” !
J. M. Diederik Kruijssen, James E. Dale and Steven N. Longmore!
http://adsabs.harvard.edu/abs/2015MNRAS.447.1059K!
!
2015, MNRAS, 447, 1059-107!

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27. “Cross-sections for planetary systems interacting with passing stars and binaries”
Gongjie Li and Fred C. Adams!
http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stv012!
!
2015, MNRAS 448, 344-363!

!

28. “3D kinematics of the near-IR HH223 outflow in L723 R.”!
R. Lopez, J. A. Acosta-Pulido, R. Estalella, G. Gomez, and B. Garcia-Lorenzo (Spain)!
http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stu2607!
!
2015, MNRAS 447, 2588-2602!

!

29. “Study of morphology and stellar content of the Galactic H II region IRAS 16148−5011!
”!
K. K. Mallick, D. K. Ojha, M. Tamura, H. Linz, M. R. Samal and S. K. Ghosh!
http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stu2584!

!

30. “SHADOWS CAST BY AWARP IN THE HD 142527 PROTOPLANETARY DISK” !
S. Marino, S. Perez, and S. Casassus!
http://stacks.iop.org/2041-8205/798/i=2/a=L44?
key=crossref.ce45c878a03a5d94cabe21b0186f7bac!
!
2015, ApJL 798, L44!

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rbital structure and evolutionary timeline

M. Diederik Kruijssen,1‹ James E. Dale2,3 and Steven N. Longmore4-Planck Institut für Astrophysik, Karl-Schwarzschild-Straße 1, D-85748 Garching, Germany
ellence Cluster ‘Universe’, Boltzmannstraße 2, D-85748 Garching, Germany
26. “The dynamical evolution of molecular clouds near the Galactic Centre – I. Orbital
versitäts-Sternwarte München, Scheinerstraße 1, D-81679 München, Germany
andJohnevolutionary
ophysics Research structure
Institute, Liverpool
Moores University,timeline”
IC2, Liverpool!Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK

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J. M. Diederik Kruijssen, James E. Dale and Steven N. Longmore!
machine-readable table with time steps of !t = 0.01 Myr is availhttp://adsabs.harvard.edu/abs/2015MNRAS.447.1059K!
pted 2014 November 27. Received 2014 November 6; in original form 2014 September 30
able in the Supporting Information accompanying this paper.
!
2015, MNRAS, 447, 1059-107!
Throughout this paper, we adopt a mean molecular weight ofµ = 2.3, implying a mean particle mass of µmH = 3.9 × 10−24 g,
and we assume a distance to the Galactic Centre of R = 8.3 kpc
ABSTRACT(Reid et al. 2014). Unless stated otherwise, all velocities are given
We recently proposed that the star-forming potential of dense
molecular
cloudsin the Central
in the reference
frameof the Galactic
Centre – from all line-of-sightvelocities
subtract
the Sun’s radial
velocity
Molecular Zone (CMZ, i.e. the central few 100 pc) of the
MilkyweWay
is intimately
linkedto towards the Galactic
Centre, which sequence
we take to beof
U⊙star-forming= 14 km s−1 (Schönrich 2012), andtheir orbital dynamics, potentially giving rise to an absolute-time
motions we subtract the Sun’s orbital motion, whichclouds. In this paper, we present an orbital model forfrom
theproper
gas stream(s)observed in the
induces a proper motion of {µl , µb } = {−6.379, −0.202} mas yr−1
CMZ. The model is obtained by integrating orbits in the inempiricallyconstrained
gravitational
Galactic coordinates (Reid &Brunthaler 2004; Reid et al. 2009).

2= 2.0) to the observed position–velocity distribution
potential and represents a good fit (χred3−3of dense (n > several10 cm ) gas, reproducing all of its key properties. The orbit is also
2 O B S E RVAT I O N A L C O N S T R A I N T S A N D
consistent with observational constraints not included inprocess,
asDtheC UtheR Rfitting
ENT O
R B I TAsuch
L MO
E L 3D
space velocities of Sgr B2 and the Arches and Quintuplet clusters. It differs from previous,
In this paper, we aim to constrain the orbital motion of the gas
parametric models in several respects: (1) the orbit isstreams
open seen
ratherthancloseddue toisolate
the only those pieces ofin theCMZ.
We therefore
twice
extended mass distribution in the CMZ, (2) its orbitalinformation
velocity that
(100–200
kmrelateds−1 ) isare directly
to theballistic orbital dynamics
of thegas. Of
goal is to connect the resulting
withthecourse,
focustheoflong-termthe (eccentric)
as high as in previous models, and (3) Sgr A∗ coincides
picture to additional constraints and physics, such as e.g. hydroorbit rather than being offset. Our orbital solution supports
the recently proposed scenario indynamics, star formation, feedback, and the 3D geometry derived
which the dust ridge between G0.253+0.016 (‘the Brick’)and
Sgr B2studies.
represents
anofabsolutefrom absorption
Aspectsthese are discussed qualitatively
time sequence of star-forming clouds, of which the condensationin Section 5. was triggered by the tidal
compression during their most recent pericentre passage. We position the clouds on a common
timeline and find that their pericentre passages occurred 2.1
0.30–0.74
ago. Given
their gas
short
GeneralMyr
propertiesof the densein the CMZ
free-fall times (tff ∼ 0.34 Myr), the quiescent cloud G0.253+0.016 and the vigorously starBefore presenting our orbital model for the dense gas in the CMZ,
forming complex Sgr B2 are separated by a single free-falltimeof evolution,
we first
discuss
the observed implying
structure andthat
the main existing model.
−3 complete
star formation proceeds rapidly once collapse has beenThe
initiated.
providethe) gas morphology within the cendense (n We> several
103 cmtral ofdegreeof the CMZ(i.e. withina galactocentric radius oforbital solution, as well as several quantitative predictions
our model(e.g. propermotions
150 pc) is shown
Fig. 1, whichof
in thered reveals a pronounced
and the positions of star formation ‘hotspots’). The paperRis∼concluded
with ina discussion
figure-eight shape. This shape inspired the twisted-ring model ofassumptions and possible caveats, as well as the positionMolinariof the modelin thewhich
Galactic
context,
et al. (2011),
is the mostrecent model for the struchighlighting its relation to large-scale gas accretion, theturedynamics
of
the
bar,
the
x
2 orbitalof the CMZ (see below).
While the morphology of the molecular gas in the CMZ provides
family, and the origin of the Arches and Quintuplet clusters.

a clear picture, its observed kinematics are intricate. The gas often

Key words: stars: formation – ISM: clouds – ISM: kinematics
andvelocity
dynamics– Galaxy:harbours multiple
components
along the line of sight (e.g.
Bally et al. 1988; Morris & Serabyn 1996). This complicates thecentre – galaxies: ISM.
dynamical analysis of the gas and obstructs a straightforward deriva-tance, a fundamental physical understanding of star formation has
N T RO D U C T I銀河系の中心周りにある分子雲ガス、CentralON
not been achieved (McKee & Ostriker 2007; Kennicutt & Evansformation is one of the fundamental physical processes driving
2012; Krumholz 2014).
aryonic evolutionMolecularof the cosmos,Zone(CMZ)の運動の軌道を観測値からfrom reionizing the Universe
Several complicating factors are to blame for our limited unry high redshift to regulating galaxy evolution and depositing
derstanding of star formation. For instance, star formation is inmodelingした。!
ls in the interstellar medium (ISM), enabling the development
herently a multiscale process, of which the physics connecting theanetary systems and eventually life. Despite its critical impordifferent scales are highly complex. A wide range of recent work
has attempted to address this problem by connecting the empiriCMZでの星形成は朝夕力による影響を受けるた
cism of galactic star formation relations to the cloud-scale physics
of star formation (e.g. Bigiel et al. 2008; Heiderman et al. 2010;
ail: kruijssen@mpa-garching.mpg.de
め軌道決定することが重要らしい。今回、決定!

14 The Authors した軌道パラメータは先行研究の値と異なるとこshed by Oxford University Press on behalf of the Royal Astronomical Society

ろがある；!

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Figure 1. Three-colour composite of the CMZ within a galactocentric
radius of R ∼ 150 pc. Red shows an integrated-intensity map of the
HOPS NH3 (1, 1) emission (see text) to indicate the gas with a density
n > several 103 cm−3 , green shows the MSX 21.3 µm image (Egan et al.
1998; Price et al. 2001), and blue shows the MSX 8.28 µm image. The
MSX data shows PAH emission (mostly tracing cloud edges), young stellar
objects, and evolved stars. The dotted line shows the model of Molinari et al.
(2011).

Orbitation oa simpthe momodel
for thgravitIn trepressymbo
5 of M
from athe bocentreorbitaC, pasfrom tBrickand thloop aThe
new incan beagreemsystempositioconstrCMZ.

2.2 SyThe coforwa
systemto obtaels to.
southeet al. 2
The
across0.◦ 8, ov
thoughnot degas. Inobservof thethe CMjssen &Highe
NH3 (1in thethe destrong(excepfrom oet al. 2
In shigh-dmolec
be traccompa
irrespe1. CMZで質量分布が広がっているため軌道は開いている

2. 軌道速度(orbital velocity, 100-200 km/s) は以前のモデルに対して２倍大きい!
3. Sgr A* は(この研究で求めた)軌道の焦点にある!
・fittingに利用していないSgr B2, Arches and Quintuplet の軌道も一貫している。!
・The Brick, G0.253+0.016の形成が、tidal compressionで誘発されたという我々の提案したシナリオをサポートする!

!

この論文はシリーズ物でこれはIで、IIではシミュレーショ
ンについて出す予定。!

Figure 6. Top-down view of our orbital model (solid line), with the observer
located in the negative-y direction. As in Figs 2 and 4, the colours referto the four coherent streams in position–velocity space. The dots indicate
the implied positions in the Galactic plane of several GMCs and cloudcomplexes in the CMZ, the plus symbols indicate pericentres, the crosses
mark apocentres, and the open black circle denotes the position of Sgr A∗ .
Note that we only show the part of the orbit that is currently associatedwith the observed gas structure – in the future, the gas stream (or its starformation products) will continue on the same rosetta-like orbit, beyond theend of Stream 1 shown here.

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Other, more fundamental open questions include the existence ofgaps in the observed gas distribution in comparison to our orbital
model, as well as the validity of the adopted gravitational potential.
These points are considered in Section 5.2.

1066

4 I M P L I C AT I O N S F O R C M Z C L O U D S

and cloud complexes in the CMZ in the context of their dynamical
history. Fig. 6 shows a top-down perspective of the orbit, along with
the implied positions in the Galactic plane of the main GMCs in
the CMZ. One of the robust conclusions of our orbital parameter
survey is that the ‘dust ridge’ sequence of GMCs between the Brickand Sgr B2 recently underwent a pericentre passage (as requiredin the scenario of Longmore et al. 2013b). In addition, the 50 and
20 km s−1 clouds are the closest to Sgr A∗ of all objects underconsideration here, but not quite as close as they were in previous
models (R < 20 pc; Molinari et al. 2011).9
Comparing Figs 6 to 4, we see that the independent streams
never approach each other closely. From the top-down perspectivein Fig. 6, there are three crossings, of Streams {1, 2}, {1, 3} and{2, 4}, at longitudes of l = {−0.◦ 8, 0◦ , 0.◦ 7}. After identifying

ude stream towards
me time gradually

onding to intervals
he latitude b, linee gas stream. Theo be the resolution, whereas those on
ond to the velocity
ncertainties can be

9 Another difference with respect to previous work is the orientation of the

Figure 2. Comparison of the Molinari et al. (2011) parametric orbital model (dotted line) with the observed integrated-intensity map ofNH3 (1,1) emission near the Galactic Centre, tracing gas with volume densities n > several 103 cm−3 (grey-scale). Symbols with error bars show thecoherent phase-space structure obtained as described in Section 2.2. We
have divided the gas into four coherent streams in position–velocity spacethat are colour-coded as indicated by the legend. In the Molinari et al. (2011)
model, the front side of the gas stream consists of Streams 1 and 2, whereas
the back side consists of Streams 3 and 4. The open black circle denotes
the position of Sgr A∗ . Top panel: distribution in Galactic longitude andlatitude {l, b}. The red arrow indicates the observed proper motion vectorof Sgr B2 and the bright yellow arrow represents the model prediction. The
latter highlights the model’s direction of motion, which is anticlockwise on
the left-hand side of the orbit and clockwise on the right-hand side. Bottom
panel: distribution in Galactic longitude and line-of-sight velocity {l, v los }.
The model’s direction of motion is anticlockwise. The red circle indicates afeature in position–velocity space that is discussed in detail in Section 2.3.

phase-space strucis collected at 113
ecessary for fitting
distribution of gas, together with theata points trace the
ページ 3/10
Fig. 1. The line-oftwo main ‘arms’, the first of which is constituted by our Streams
en seen from above
2–4, whereas the second one represents our Stream 1. In addition,
the Galactic longithey distinguish two more arms that are offset to slightly higher
longitudes (part oflongitudes (and in one case latitudes) from Sgr A∗ . These werefrom our position,

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on February 12, 2015

ngle velocity comtitude of the pixel
iple velocity comatitude of the pixel
rresponding to thehis choice is madeng the line of sight,
ission from the gasached at a different

these positions in ththese the streams ar!z > 20 pc. This isorbit – the ratios becillation periods are
regularly intersect w
must also be a time
However, their total
indicating that suchtidally stripped or fe
act with other stream
GMCs during theiron Stream 1.
We proposed in L
tre passage of the St
Sgr B2) caused themleading to an acceler
ally gravitational cois that the GMCs oquence of contractio
each cloud’s evolutio
passage. While theturbulent dissipation
simulations, we cantime since pericentre
absolute timeline on
the scenario of Long
Table 2 lists theof the objects in Fi
last (next) pericentre
tively). We see that t!tp,last = 0.30 Myr aapocentre passage an
Table 2 lists the time
This is done because
pericentre can be suthat the time differen
better. We find that t
is independent
generally constrai
associatedindustridge (i.e. frombetweenthe Brick–an
our model
The star formatiofer(i.e.
moreStream– the Bric2(Kauffmannet al. 20
position–velostar-forming protocl
line of sight (the context of the Lo
tooncethecollapse
gas thathatisformationand is proceeds
likely p
This is twice as fas
unlesset here,al. (2011)
modeliabout
onepossibil
free-fall tiThisn ∼ 104 cm−3 (Longthe Arches anthis model, the Brickhavebeen asstimeof evolution.
are s
1Clouds(see d/e/f
Sectionand Sgr B2. BasedJ. M. D. Kruijssen, J. E. Dale and S. N. Longmore

Cloud evolution and the physics of star formation
After; This4.1
work
The orbital solution of Section 3.2 allows us to consider the GMCs

gas is mapped as

Brick towards Sgritudes towards low
ssion towards hightion of Sgr C.

Orbital structure!

ale and S. N. Longmore
Before; Molinari et al. 2011のモデル

A∗ (at the locations
Bally et al. 2010),
three independentof these connect ind it is not possible
or a chance projecnd of a gas streamle), suggesting thatrojection of an inw the brighter of the(the orbit is open)!

orbit. In the papers by Molinari et al. (2011) and Johnston et al. (2014), aclosed, elliptical orbit is rotated around the orbital rotation axis such that
it is tilted with respect to the line of sight (i.e. the apocentres occur awayfrom y = 0 pc). As is shown in Fig. 6, our orbital model exhibits little
such rotation, with both apocentres close to y = 0 pc. This difference arises
because the orbital velocity varies in our model, allowing a good fit at all
longitudes without the need of boosting the line-of-sight velocity by rotating
the model.

ically associa(visible in gr
stripped.
This wouldbstructuresurrof the cloud’s outer laresultfieldfromamagnetic
strengt
2014;
et al.
et Rathborneal. 2014),
is not necess
they are caus
components.7

10 While cloud collaps

In additionis dynamicaleral physicalmodels.
Figure 4. Comparison of our orbital model (solid line) with the observed
integrated-intensity map of NH3 (1, 1) emission near the Galactic Centre,
tracing gas with volume densities n > several 103 cm−3 (grey-scale). Symbols with error bars show the coherent phase-space structure (see Section 2.2)
to which the model was fitted. We have divided the gas into four coherent
streams in position–velocity space that are colour-coded as indicated by thelegend. In our model, the back side of the gas stream consists of Streams
3 and 4, whereas Streams 1 and 2 represent the two (independent) ends ofthe stream on the front side. The open black circle denotes the position of
Sgr A∗ . The model starts at Stream 2 (the overlap with Stream 4 is coincidental but could fit too) and continues through Streams 3 and 4 to Stream 1.
Top panel: distribution in Galactic longitude and latitude {l, b}. The redarrow indicates the observed proper motion vector of Sgr B2 and the brightyellow arrow represents the model prediction. Bottom panel: distribution in
Galactic longitude and line-of-sight velocity {l, v los }.

(i) The ext
bit that is op
azimuthal, anture can surv
disrupted by(ii) The no
ranging fromat pericentre.
As a result,shorter than{PR,mol , Pφ,m(iii) The b
the position S
argument to
the fact that t
20 km s−1 clo
model. In ou
leads to the onecessary to

7 A peak of sh

has been put fcloud–cloud cod binaries

ngjie Li1 and Fred C. Adams2,3‹

vard–Smithsonian Center for Astrophysics, Institute for Theory and Computation, 60 Garden Street, Cambridge, MA 02138, USA
ics Department, University of Michigan, Ann Arbor, MI 48109, USA“Cross-sections
forMIplanetary
onomy Department,27.
Universityof Michigan, Ann Arbor,48109, USA systems interacting with passing stars and binaries”

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ABSTRACT

Most planetary systems are formed within stellar clusters, and these environments can shape
their properties. This paper considers scattering encounters between solar systems and passing
cluster members, and calculates the corresponding interaction cross-sections. The target solar
systems are generally assumed to have four giant planets, with a variety of starting states,
including circular orbits with the semimajor axes of our planets, a more compact configuration,
an ultracompact state with multiple mean motion resonances, and systems with massive
planets. We then consider the effects of varying the cluster velocity dispersion, the relative
importance of binaries versus single stars, different stellar host masses, and finite starting
eccentricities of the planetary orbits. For each state of the initial system, we perform an
ensemble of numerical scattering experiments and determine the cross-sections for eccentricity
increase, inclination angle increase, planet ejection, and capture. This paper reports results fromover 2 million individual scattering simulations. Using supporting analytic considerations, andfitting functions to the numerical results, we find a universal formula that gives the crosssections as a function of stellar host mass, cluster velocity dispersion, starting planetary orbital
radius, and final eccentricity. The resulting cross-sections can be used in a wide variety of
applications. As one example, we revisit constraints on the birth aggregate of our Solar system
due to dynamical scattering and find N ! 104 (consistent with previous estimates).
Key words: planets and satellites: dynamical evolution and stability – planetary systems.

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Gongjie Li and Fred C. Adams!
http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stv012!
pted 2015 January 4. Received 2014 December 15; in original form 2014 August 30!
2015, MNRAS 448, 344-363!

太陽系のような惑星系がクラスター内の他の恒星と遭遇したときの惑星軌道の変化をシ
the dynamical constraints derived for the birth aggregate of the So-

N T RO D U C T I O N

lar system depend on many variables, including assumptions madeミュレーションで調べた。数値計算の結果から、cross-sectionを中心星質量、クラスター

ge fraction of planetary systems form within stellar clusters
about the cluster properties, any other constraints imposed on thea & Lada 2003;速度分散、初期の惑星軌道半径、終状態での離心率の関数として求めた。!
Porras et al. 2003) and these birth environproblem, and the interaction cross-sections.
s can influence their resulting properties (e.g. see the reviews
This present study focuses on the cross-sections themselves, anddams 2010; Pfalzner 2013). One potentially important process
expands previous work to include a much wider range of parameter
Motivation;
rs when binary systems– and single 惑星系の多くはクラスター内で形成される。近年の研究で、太陽系の惑星はstars – fly past solar sysspace; the implications for the solar birth environment are thenand disrupt the orbits of their constituent planets. This typebriefly considered at the end of the paper. For studies concerningよりコンパクトな配列(軌道)で形成され外側に軌道が移動したと可能性が示唆されていattering interaction has been studied in the field (Laughlin &our Solar system, most previous work has calculated the crossms 2000), and withinyoungembeddedclusters
(e.g.
Adams
sections for this mode of disruption by considering the initial orbits
る。この研究では近接遭遇のcoros-sectionに着目して調べる。!
2006; Malmberg et al. 2007; Malmberg, Davies & Heggieof the giant planets to have their present-day values of semimajor; Boley, Payne & Ford 2012; Chatterjee et al. 2012; Dukes
axis. However, some recent work suggests that our Solar systemumholz 2012; Hao, Kouwenhoven & Spurzem 2013; Pacucci,
may have begun in a more compact configuration (Gomes et al.
method; 数値計算。（試行回数200万）。
この研究では、太陽系likeな系；木星、土ra & D’Onghia 2013), where the latter results can be used to2005; Tsiganis et al. 2005), and the planets may not have reachedde constraints on
the possible birth environment of our own
their present-day orbits until the Solar system reached an age of星、天王星、海王星の４つの巨大惑星をもつ惑星系をターゲットとして基本的には考える
system (e.g. Adams & Laughlin 2001; Hester et al. 2004;
hundreds of millions of years. One motivation for this present study(現在の太陽系、compact、離心率が異なるものとかを初期の惑星系としておく,Table1)。
ams & Gaidos 2007;
Spurzem et al. 2009; Portegies Zwart
is thus to determine cross-sections for Solar system disruption for; Adams 2010; Williams 2010; Pfalzner 2013). We stress that
more compact configurations. Note that the sign of the effect is not
これにbinary(もしくはsingle star)が通過する。!
obvious a priori: the geometrical cross-section of the compact Solarsystem is smaller, and hence implies a smaller interaction cross346
G. Li and F. C. Adams
section. However, the decreased relative separations of the planets
ail: fca@umich.edu!
!

ページ 4/10

Table 1. Summary of Solar system configurations. In the standard configuration (first⃝
line), theC 2015 The Authors
planets have the same masses
andsemimajor
axes
as
those
in
our
Solar
system
but
start
with
zero Society
Published by Oxford University Press on behalf of the Royal Astronomical
eccentricity. For the other configurations, the table entries list the initial values of the parametersthat are different from those of the standard configuration (so that all of the unlisted parameters
have their standard values).

Configuration

Jupiter

Solar system architectures
SaturnUranus

NeptuneStandard
Compact
Resonant
Eccentric # 1
Eccentric # 2
Massivee=0
a = 5.20 au
a = 5.88 au
e = 0.049
e = 0.10
mP = 1mJ

e=0
a = 8.67 au
a = 7.89 au
e = 0.057
e = 0.10
mP = 1mJ

e=0
a = 14.4 au
a = 10.38 au
e = 0.045
e = 0.10
mP = 1mJ

e=0
a = 24.1 au
a = 12.01 au
e = 0.011
e = 0.10
mP = 1mJ

where a is sampled uniformly. If every encounter within this area
leads to the ejection of Neptune, and all encounters outside thisarea (which are not computed) have no effect, then fD = 1; theon the binary centre of mass (corresponding to the choice B = 100
in equation 2).
Using a Monte Carlo scheme to select the input parameters ac-

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↓Cross-section vs inclination (summation) . inclinationは初期ではどれ↓Cross-section vs 離心率も同じ公転面からスタートして、図は４線は木星、土星、天王星、海王星
つの惑星のinclinationを足したもの。実Blueは今の太陽系モデル、点線はこのFigure1ではCross-sections for planetary scattering 347
線、点線は衝突速度が1km/s - 16km/s
compact
配置。Figure2以降、他モデルを図示。
350
G.
Li and F. C. Adams
でstepはlog scaleでsqrt(2)倍。

!

Cross-sections for planetary scattering

355

Results

Figure 14. Scaled cross-sections versus eccentricity increase !e (equiva-

Figure 13. Scaled cross-sections versus eccentricity increase !e (equiva-

!
!

w

A
D w
m

F

O

m
m

S

Fig
!i
the
are
The
blu
dev
tex

N
A

w the
m

ity
m
(8)
fit
dev
of=
the
∼
Ho
mo
m
sm

O

m

fi
A

N

⟨σ ⟩0 exp [b0 (1 − sin !i)]m,⟨σ ⟩i =
考えると、N<10^4である
先行研究の m (8)mOO
masses and semimajor axes of our Solar system. Cross-sections are scaled
m
MNRAS
6/5
m
m
m
by the factor vb /a. The individual cases are shown as light dotted curves,
見積もりと一致する
MNRAS 448, 344–363(2015) then the
cross-section for increasing&m 。!i can be fitted using the pawhich include the four giant planets and four values of velocitymdispersion
moframeters b ≈m 3/4 and σ 0 ≈ 166 000 au2 . Note mthat the valueof the background cluster: v b = 1, 2, 4, and 8 km s−1 . The heavy solid
m bluemm the index b used here somewhat smallermthan that neededm to fit
curve depicts the average, where the error bars depict the standard deviation.
m
The red straight line shows the result for cross-sections with an exponential&m mdependence on eccentricity.
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angles,nearlycross-

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find
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the TheKozai
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fitting
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bysemimajor
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lently, post-encounter eccentricity e) for a collection of different velocity
Even in the reduced case where we consider one planet at time,lently, the post-encounter eccentricity e) for the four giant planets in eachdispersions for the background. The starting state is taken to have four giof the starting architectures for the Solar system. The individual cases are
the interactions considered in this paper involve four bodies (theant Figure
planets14.
withthe current
semimajor
axes.
Cross-sections
by
shown
as
light
dottedcurves.
The
heavy
solid
blue
curve
depicts
theaverScaled
cross-sections
versuseccentricity
increaseare!escaled(equivaFigure 13. Scaled cross-sections versus eccentricity increase !e (equivahost star, the planet, and two binary members). Unfortunately, four7/5
vb lently,/a (seetext). The individual
shown as oflight
dottedvelocity
curves,age, wherethetheerror
bars depict thestandard deviation.
Thegiant
straight
red line
post-encountereccentricitycasese) forarea collection
different
lently,post-encountereccentricitye) for the four
planetsin eachbody interactions are rather difficult to describe analytically to any−1 , equally
withdispersions
include curves
each of theplanets
for
v
=
1–16
km
s
shows thefor cross-sections
a purely
exponential
dependence
on arefor thefor
background.
The
starting
state
is
taken
to
have
four
gib
of resultthe startingarchitectures withfor theSolar system.
The individual
cases√
reasonable degree of approximation. As a result, the goal of thisspacedlogarithmically(by factorsof 2).
TheCross-sections
heavy solid blue
curve bydeant planetswith the current
semimajor
axes.
are scaled
eccentricity.
shown as light dotted curves. The heavy solid blue curve depicts the aver7/5
picts
where
error bars
depictstandard
deviation.
The redsection is relatively modest: instead of building complicated anvbthe/aaverage,(see text).
Thetheindividual
cases
aretheshown
as light
dotted curves,age, where the error bars depict the standard deviation. The straight red line
line shows
theforresultcross-sections
a purely
exponential
with include
curveseachforof theplanets
for vwithkm s−1, equally
thein
resultfor 13cross-sections
withstraighta purely exponential
dependence
The showscurvesFig.
are nearlylines on thesemi- on straightb = 1–16
alytical models for four-body (and higher N-body) dynamics, we√spaced logarithmically(by factors of 2). The heavy solid blue curve dedependence
on eccentricity.
eccentricity.
logarithmic
plot, so that the dependence of the cross-sections on
consider here basic physical principles that can be used as motivapicts the average, where the error bars depict the standard deviation. The red
eccentricity is nearly exponential. For purposes of illustration, we
tion
combine these heuristic results withstraight
line
shows
the
result fordeviation.
cross-sections
a purely
exponential
354for scaling
G. Lilaws.
andWe
F. thenC. Adams
The curves in Fig. 13 are nearly straight lines on the semi- error
bars depict one standard
Thewithheavy
straight
red line
use an exponential fitting function of the form
our detailed numerical determinations of the cross-sections. The
dependence on eccentricity.
logarithmic plot, so that the dependence of the cross-sections on represents
the same exponential dependence given in equation (7)⟨σ⟩resultisphysicallymotivated
fitting
formulathatcharacterizethee eccentricity is nearly exponential. For purposes of illustration, we
Even in the reduced case where we consider one planet at time,and used in Fig. 13. The rms of the relative differences between= απ ℓ exp[ b (1 − e) ],(7)
error bars depict one standard deviation. The heavy straight red line
use an exponential fitting function of the form
, e).
cross-sections
the parameterinterest(a, bodies
v b , M∗ (theathe interactionsoverconsideredin this space
paper ofinvolvefour
therepresents
numericallycross-sections
andgiven
the incurve
given(7)bythe determinedsame exponential
dependence
equation
Tostar,startthethe
discussion,
the simplest
case wherehostplanet,and two consider
binary members).
Unfortunately,
four-the⟨σ ⟩first
where thefactor enforces consistency with the ejection crossequation(7) inis ∼13
per Thecent.rms
Theofcross-section
curves are thus
selfe
and used
Fig. 13.
the relative differences
between= απ ℓ exp[ b (1 − e) ],(7)2cross-section
for interactions
is the togeometricalcross-section
πaCross-sectionsforplanetary
scattering
357
sections considered
above. For the value b = 4/3, we obtain a good
similar
to
this
level
of
accuracy.
Furthermore,thedependence
body interactions
are rather difficult
describe analyticallyto anyathe numerically determined cross-sections and the curve given byof
fit to thecalculated,cross-sections,as shown
byejection
the heavycross-sections
on pervelocity
dispersion
is nearly
independent
provideda planetin its initial orbit.
wegoalconsiderreasonablebydegreeof approximation.
As aFurther,result, theof thisthewherethe first scaled
factor enforcesconsistency
with thecross- theequation(7) is ∼13
cent. The
cross-section
curvesare thus self-of
Figure 1. Cross-sections for eccentricity increase for the current Solar
Figure 2. Cross-sectionsforeccentricity
increaseforthecurrentSolar
redlineassociationin Fig. considered
13. between
Exceptabove.
for
first
(e =variables
thewe
exponential
thesimilar
dependence
starting
semimajor
axis a of theThechangesin valuethesin!i andesection tois berelatively
modest:insteadof duringbuilding
planets
independentof eachother
thecomplicatedencounters.anThis
sections
Forpoint
theb0.1),= 4/3,
obtaina goodto thisonlevelof accuracy.
Furthermore,
the planet.
dependence ofsystem architecture and for a more compact configuration motivated by thesystem architecture and for a resonant configuration motivated by the Nicefit(straight
red
line)
agreeswiththeaveragevalues(solidblue
Thedependenceoftheinteraction
cross-sectionson
the massofof
providesan
intriguing
topic
for
additionalwork.
To
leadingorder,fittothecalculated,scaledcross-sections,
as
shown
by
the
heavy
the
cross-sectionsonvelocity
dispersion
isnearlyindependent
alyticalmodelsforfour-body(andhigherN-body)dynamics,
wecross-section
willbysystemgravitational
wecanNice model.
Forbetheenhanced
current Solar(solid bluefocusing,curves), theso
four
giant
model. For the current Solar system (solid blue curves), the four giant
curve)
tocurrent
withinabout
per cent,i.e.
thedifference
isthe
much
less
thethehoststar is somewhat
morecomplicatedforplanet.
the other varilineactions
inFig.
13.3axes
Except
for
first
point
(e = 0.1),
exponential
dependence
on starting
semimajor
axis athanof thecanonical
writtenin terms
oftheorbitalelements
haveconsiderhere
thatcross-section
canusedmotivawrite
down
anbasic
heuristic
thein
the form
planets
arestartedphysicalwithexpression
theirprinciples
current for
semimajor
axesbeandzeroaseccentricity.
planets are started thewiththeirredsemimajor
andzeroeccentricity.
fit width(straight
red
line)
agreesthe(soliddependenceinofFig.
the 4.
interaction
cross-sections
theincreases,
mass of
star
than
the
the
distributions
asare
measuredbyvalues
theangles
standard
as illustrated
As the massM∗ of theon
Figure
7. forms
Cross-sections
forincreasing
thewithspreadofaverage
inclinationof blue ables,Thethetion forWe
theseheuristic
results
with where For
For scaling
thecompact
configuration(dashedred
curves),the planets
startedconfiguration(dashedofred
curves),the planets
started!laws.
"then combine
Figure6. Cross-sections
for eccentricity
increase
inare
systemsthethe resonant2
vesc
curve)
to10.38,
within
about
3au
cent,
i.e. thedifferenceismuchhost star is somewhat
more
for the othervaricross-sections,considered
ascomplicatedfunctions ofthan
eccentricity,
becomedeviations.
Another
measureofper
thequalityofthethesamefit isplane;
provided
the axes
planetary
orbits.
Alloftheandgiant
planets
arestartedinthe less thethewith giant
semimajor
axes
havinga fixedmratio
aj+1with semimajor
a = 15.88,
7.89,
12.01
(for
the
analogues
where
Jupiter(jThe=cross-sections1)
j = 5/3,
planetsallhave(dashedred
curves).
The
our⟩0 detailed
determinationsofJ /athecross-sections.
P = 1m≈απ
a 2numerical
1+
,mass
(5)
⟨σ
2
2
and
Zbetween∝(i/2).
(9)
ethan
"the
∝
of thedifferent
star increases,
the
width
of
thesindistributions
asthe
measuredby afterthecrossstandard
ables, The
as illustrated
inFig.
4.curves
As theinmass
M4∗ (for2 location aJ = 5.2 au. For both sets of cross-sections,steeper.
spacingoftheFig.
stellarrelativedifferencesthenumericallydeterminedofJupiterto
Neptune).
For
both
sets
of
cross-sections,thecurves,fromisstartedatitspresent
quantity
!i
is
the
total
range
of
inclination
anglesoffourorbits
the
v
for
the
current
Solarsystemarchitecture
are
shown
for
comparison(solidresult is physically motivatedfitting formula that characterize the∞
2 deviations. Another measure of the quality of the fit is provided
thecross-sections,consideredasfunctionsofeccentricity,
become
top to bottom,
correspondto Jupiter(bottom),
Uranus,and the
Neptunethe curves,from top
to bottom,
correspondto Jupiter(bottom),
Saturn, semimajor
masses) grows with e, so that the curves are not self-similar. In
sections
used inconstructing
andform given
encounters.
Cross-sections
are Saturn,shownFig.
for 13a varietyofexponential
velocity dispersions,blue curves).
In bothcases, theplanets
arestartedM∗ ,currente). thecross-sections
theparameter
spaceof interest(a,withv b , theapparent
relation
betweentwovariables(asobservedinis overtheescapetarget
(at
where
v esc
−1
steeper.
The
spacing of thein Fig.
4 (fordifferent stellar
thes−1
relativedifferences
thesthenumericallydetermined(top).
Since12.
theorbits
start
with
zero
eccentricity,theeccentricity
increase
Uranus,and
Neptune(top). speed
Since thefrom
orbits thestart withzerosystemeccentricity,
the
from
vThe
1 km
(top
curve)
tov bbetween
=the
16 km
(bottom
curve),
wherecrossspiteof thiscomplication,
wecurves
can still
fit thecross-sections
with
by
(7);
the
rms
of
these
relative
errors
is √
∼12
per
cent.
FigureCross-sectionsforplanetary
escapeversusstarting
semib =equationaxes
the giantplanetsin ourSolarsystemandwith zeroeccentricity.
Toeccentricity
start
theofincrease
discussion,considercase
wherethebesimulation
results)
could
thusofFig.
be(byevidence
ofexponentialan2).equipartition!e = e, where
isthe
the Next
post
encountereccentricity.
!e
= e,
where
e isthethe simplest
post encountereccentricity.
masses) grows
withofe, stellar
so thatmass,
the curves
are thenot accuracy
self-similar.
In
sections
used
inlogarithmicallyconstructing
13fourand giant
theof
form
given a power-law
isfortheasymptotic
relative
speed
locationoftheplanet),andv∞
theeThevaluesareevenlyspacedfactorsfunction
althoughof theweonconsidertheeffects
velocity
dispersionof the
major
axis.
12
pointstheplotcorrespondtotheplanets
2
Cross-sections
are
shown
analoguesof
Jupiter(bottom
curves),Saturn,cross-sectionforinteractions
isthegeometricalcross-section
π
a
spite
of
this
complication,
we
can
still
fit
the
cross-sections
withbyequation
(7);
the
rms
of
these
relative
errors
is
∼12
per
cent.
like
mixingoftheactions(see
Lichtenberg&
Lieberman
1992).
14. Scaled
cross-sections
versuseccentricity
increase!e (equivatween thetwosystems.
In ordertoversus
pass within
this cross-sectional
approximation is not expected to be as high as in the previous cases.
clusterenvironment.
Asarchitecture.
shownin Fig.
3 inthe previous
Figure13.
Scaledeccentricity
increase !e (equivain each of Figure
threebackgroundversionsof theinitialSolar system
The
symUranus,
andcross-sections
Neptune(top curves).
providedby a planet
in itsinitialorbit.beFurther,
we considerthea power-law function of stellar mass, although −µ
the accuracy of the
Next wethe
considertheeffects
ofapaper,thevelocity
dispersion
ofbetheAlthoughbeyond
scope
of
this
paper
this
issue
shouldarea,
theinteracting
star(binary)
must
about
the
same
distance
lently,post-encountereccentricitye)forcollectionofdifferent
velocity
We thus consider a scaling of the form ⟨σ ⟩ ∝ M∗ , and vary the insection,
thecross-sections
vary successive
withthe post-encountereccentricity
bolsofrepresent
starting
the semimajoraxes of thesection).
The results areshownotheras theblueencounters.
curvesin Fig.
orbit
JupiteraJ = 5.2
au,andstates,
thenletsufficiently
eachplanetlently,thepost-encountere)solid
for thefour
giant
planets
Solaratdifferent
systemisunstableonincluding
time-scales
planets
to be
independent
ofeccentricity
eachduringthe
This1, in eachapproximation is not expected to be as high as in the previous cases.
backgroundclusterenvironment.long
As shown
in Fig. 3(Batygin
in the previous
exploredfurther.
fromtheplanetas its thehostcross-sections
star,thesoSolar
thatforsystem.
theplanethas
astartingchancedex µ to find the best fit. The choice µ = 1/3 provides
the lowest rms
withapproximatelythefunctional
form
over
wide rangeof four gipresent-day
Solarsystem(stars),a compact
configuration
with
5/3
for
background.
The
starting
state
isa taken
to havealsoarchitectures
presents
a morecompacthavea semimajor
axis
that
isthe
largerthan
thepreviousonebya semimaof
thewhichstarting
forThe
individual&dispersions
Laughlin2008;
Laskar& sameGastineau
We thus consider a scaling of the form ⟨σ ⟩ ∝ M∗−µ , and vary the insection,
the across-sections
vary2009).
withthepost-encountereccentricity
cross-section
will be enhanced
by gravitational
focusing,
so we can cases areWecanextract
potentially
importantcross-section
from
these
jor
axis
ratios(open
triangles),andanultracompact
Solarsystemstarting
in
.
Only
the
leadingcoefficientchanges.
Moreover,theuniformoftherelative error. Fig. 15 shows the result by plotting the scaled
v
of
being
ejectedfromitsoriginal
Solarsystem.
This
expression
configuration(described
below).
The
error
bars
(not
shown)
due
factorof
5/3.
This
evenly-spacedSolarsystemthus
extends
out
to
b
ant
planetswiththecurrent
semimajor
axes.
Cross-sections
arescaledbyshown
asconsiders
lightdottedcurves.
Theforheavy
solid blue
curve
dex µ to find the best1/3fit. The choice µ = 1/3 provides the lowest rms
withapproximately
the samefunctional
form
over
a readilywide rangeofAnother
way
in
which
planetary
orbitscanbealteredbyscattering
post-encountereccentricities
sufficiently
larger thethanaverthe
write down
an heuristic
expression
the cross-sectionin thedepicts
form
results.
The
scatteringinteractions
consideredhere
canin7/5 results,
multiple(open
squares).
The
red
line
shows
the
cross-section/a 15(againincludingspacingof
thein
curvesFig.
3casesindicates
that
theascross-sections
cross-sections⟩M∗ Fig.
to incompleteCarlosampling correspondto relativeonly
24 au.MMR
Fig.
1,insolid
indicatethatthecrossthus where
represents
theMonte
escapecross-section.
The
parameter
α straightiserrorsa di- red line
. Onlytheleadingcoefficientchanges.
Moreover,uniformof the relative⟨σerror.
shows
the resultthe
by linearplottingdependence
the scaled
vshownvThe/a(seeis
The
are
shown
light
curves,btext).
age,
the!errorbars
the standard
deviation.
The
encountersbychanging
their
inclination
angles.
Forallintheofadotted
the"depictb thevalues).
creasethespreadof individual
inclination
anglesofSolar
2indicated
limiting
form
ofofequation(6);intheFig.
blue
dashedcurvebodies
shows
on semimajor axis
a) as
must
have
a power-law
dependence
theoutervelocity
dispersionv
1/3 a function of post-encounter eccentricity.
withstartinga root-mean-square
(rms)
value
of is
∼4.4
per cent.
sections
for bythecompact
configurationaresomewhatsmallerthanvesc
mensionlessof orderunity
andincluded
to encapsulate
theeachcurves3onindicates
thecross-sections⟨σ ⟩M
2constantwithincludespacing
curvesforofthe
thescattering
planets
=
1–16
km(so
s−1b , equallycross-sections∗ /a (again including the linear dependence
shows
with
aof
purelyexponential
dependenceonsimulations,
we
start
the
four
giant
planets
infor
thev bthat
same
plane
we
considereffects
planetary
mass onthe
scattering
απresultaNext
1for+ cross-sections, thethis(5)⟨σ ⟩0 ≈thesystem.
On
the
other
hand,
eventsthemselves
have
w
m
m
σ ∝
√
the
full
form.
(to
leadingorder).
We
have
exploredscalings
withvelocity
deThe
light
dotted
lines
show
the
individual(scaled)cross-sections
In
Fig.
1,
and
throughout
paper,
thecross-sections
forinthoseobtainedwiththecurrent
semimajor
axes.
For
this
compact2
semimajor
axis a) as
a functionof post-encounter eccentricity.
must have a power-law
dependence
on theheavy
velocity
dispersionv b 356
the uncertainties inherent
v∞ in this approximation. After inserting the−γ
spacedlogarithmicallyfactors
ofand
2).
solid
blue
de-ontheG.dottedLi
F.shows
C.thetheAdams=
ieffect
iof=
iNform=(by0).
theencounters,
thebestinclinationthateccentricity.
cross-sections.
The toresults
shown inallFig.
6 forthethatusual Solar
Jrms
S =
U
on
planets
intightorbits,
such
asThe
the
multiplanet
systemsfor(notplanetary
scattering
357
find
that
the
occurspendencethe⟨σ⟩After
∝
vhave
heavy
blueandcurveshow
average.(scaled)
The error
bars depict
creasing
the eccentricity
e = 1 areincorporateof theways
SolarCross-sections
system,theilittle
errorsshown)
duetoincomplete
sampling
Cross-sectionsforplanetary
scattering
355curve(toleadingorder).
We
exploredscalings
withfitvelocity
de- andF
The light
lines
individualcross-sectionsb
expressionforescapespeed,
we
obtainpicts
theaverage,
where
the
error
bars
depictthestandard
deviation.
The
red
−γ
anglesofthefourplanetsare,ingeneral,non-zero.
Wedefinethe
Cross-sectionsforplanetary
scattering357
is thetheescape
speed
from
the
targetsystem(at thewherethevplanet
observedbytheKeplermission(Batalhaetal.
2013).
However,thesystemparameters
andfor
anSolaranalogueSolarsystemwhere allFigureoftoisolatethedependenceofthecross-sectionson
initialsemimaF
m
m
esc
can
be
removedfromits
system.
These
channels
are
∼4.6
per
cent.
for γ ≈
7/5.
Using
this
choicepower-lawindex,
plot
corresponding
standard
deviation,
which islargertheand findsemithat thewe
best
fit theoccurs theandpendenceof theescapeform⟨σ ⟩ ∝ofvthe
the heavy blue
curve shows
the average.
Theerror thanbars fordepict12. Cross-sectionsforplanetary
versus!
"b startingstraight
lineintheresultforcross-sections
withalong-term
purelyexponential
The
association
betweenchangesinvariables
sin
!ion
and
e classes
post-encounterspread!ioftheinclination
anglesaccordingto theistheasymptotic
relative
speed
belocationofplanet),andv ∞ are
the
outer
Solar
systemcanhaveimportantef-the cases
GM
=
1m
Both
thethe
giant
the
mass
ofthe
Jupiterm
include(1)
actually
increasingtheeccentricity
to (m
e≥
1,
which
inNextweTheconsideranevencompact
orbitalconfiguration,The
curvesinplanets
Fig.
13
nearlystraight
thesemijoraxis
fromplanet–planet
scattering
effects,
thislattercaseusesscaledcross-sections
versuspost-encountereccentricity
in we
Fig.
14,
previously
Figs
13
for
γ more
≈
7/5.
Using
this
ofgiant
power-lawindex,
plot
the consideredcorresponding
standardwhichisca
largerfor 14).
the361∗ havePlinesJ ).
axis.
12bodies
pointsonshows
theplotcorrespondto choicethefourplanetsCoec
on (compare
odeviation,p aneFig.
a 15y withe thanngand
απtwo
a 2 an1hyperbolic
+ The
→ for
απpassℓa,(6) sin major⟨σ⟩0 =association
betweenchanges
inintersectthe(thevariables!iandmotivatede threeprovidesintriguing
topic
additionalwork.
To
leadingorder,
2 withtween
thesystems.
In
order
to
withinthiscross-sectional
fects
the
inner
bodiesprovided
that
they
are
scatteredintoinorbits
expression
dependenceoneccentricity.
cludesorbitsandplanetary
orbits
that
the
again
by
theonofNicemodel,
where
the
four
giant
planets
where
weincludethelineara-dependence
found
previously(i.e. 14, Thecases
heavy
red straightline shows
the sameas before
T 15result
scaledcross-sections
versuspost-encountereccentricity
Fig.
consideredpreviously(compareFig.
with Figs
13 and(from14).
of bothsystems
start
the
same
semimajor
axes
present-day
smallerplanetmasses(byafactor
of
10),
so
they
act
more
like
test
in
each
of
versions
theinitialSolarsystemarchitecture.
The
symlogarithmic
plot,
so
the
dependenceofthecross-sections
on
avthatb
7/5
providesan
intriguing
topic
for
additionalwork.
To
leadingorder,
thecanonical
actions
written
in
terms
of
the
orbitalelements
have
"
#$
!
star, (2)inejection
from theSolarsystemthe ki-shows bols
aremasses;
in mutual
MMR.
Instarting
this
case,
choosestarting
semimajor
withsufficiently
highinclination
angles.
Morespecifically,
if the(i.e. equationarea, hostthe values
interactingstarsystem)(binary)must
beaboutbytheincreasing
same
distance
wewe
includethelineara-dependence
found
Thewheavy
red straight
linebetween
shows thethe
same
resultSas before
(from
/a).
Each
light
curve
in
theaxesfigureshowspreviously
the result⟨σ ⟩v7). Therms error
exponential
line andO thestates,includingsemimajor
ofcompact
the
our
and
zero
eccentricity.
The
figurethat
todifferent
explorewider
range
indotteda,thethe
we
also
take
this
b awhereeccentricity
isreplacednearlyexponential.
For
purposesofillustration,
we represent
w −1
m →7/5 J j ·mJm
k light dotted◦ σ ∝canonical
actions
terms
oforbital elements
have
with
the
velocity
where
wethe
have
theforasymptotic
speed
vin∞(3)theformsenergyso thethathost
theorbit
issounbound,capturebyaxes
to havevalues
of
a⟨σ
=
5.88
au
(Jupiter),
7.89
auare(Saturn),
/a).
Each
curve
in semimathe than
figureshows
theresult numerically⟩vanglesThe rms error
betweenline
and Thisthe2,straightthen
inclination
of
the
outer
orbitslarger39.
from netictheplanetasits
star,
thatwrittentheplanethas
athechanceforoneplanetandone
choiceof velocity
dispersion.
The
heavy
cross-sections
about 20mper
cent.
Solar
(stars),a factorcompactwith
5/3
m theis exponentialba, compared(4)the!i≡systemmaxcoscross-sections
increasing
theandof oneNeptune present-day
andsystemtobesmallerbyofconfiguration
1.35
to
that
considerror
depictone
deviation.
The
heavy
red lineequation 7).determined
Fbars
T 20 per cent. Thisuseanfittingfunctionofsystem.
theareeccentricities
form
the
forms
Jultracompact
Jstandardthecluster(orotherbackgroundstellarsystem),dispersion
for oneplanetone
choiceofvelocity
dispersion.
The
heavy larger
numericallydeterminedcross-sections
istheabout
1v
jthe
kand
of exponential
the
(two)
stars.
These
channelsnotmutually
exclu10.38
au(Uranus),
andcurve
12.01
au(Neptune).
WithSolar
thesesemimajor
b2 of passing
jor
axis
ratios(open
triangles),andansystemstarting
inKozai
effectcanoperate(Kozai1962;
Lidov1962),
and
the
inner
blue
shows
average
over
all
of
the
curves,
where
the
error
measureresultsfromfitting
cross-sections
withaof being
ejectedfromitsoriginal
This
expression
Figure17.
Scaledcross-sections
versuseccentricity
increase!e(equiv2 Solar
Uranusarelargely
unaffectedbytheincreaseinplanetarymass,ered in theprevious section.
The exponential
error bars
delineate
the uncertainty
and 1 Z ∝ sin (i/2).
(9)2
"∝ e
F
m
m
T
represents
thesamedependence
givenin
equation
(7)
blue
curve
shows
the
average
over
all
of
the
curves,wherethelargererror
measureresultsfromfitting
thecross-sections
with
sive,
but
the
simulations
oneofaxes, Jupiter
Saturnarea Solar
3:2
Saturnandexcited
Uranusare the age of the sysand
havedefinedthe corresponding
length-scale≡GMmultiple
MMRand(open
The
redMMR,
solid
line
shows
the
cross-section
2 the2are stopped after
2 theseℓevents
alently, post-encounter eccentricity e) for solar systems interacting awith m
portionsof inthesystemcanbeover∗ /vb larger.
thus
represents
theescapeTheparameter
α isatakesdiangular
momentum
vectorsof crossthe planetary
where
thesquares).
JMontebut
cross-sections
forandSaturnsomewhat⟨σ⟩ewe
andJupiterZ∝sin(i/2). are(9)" ∝ ecross-section.
due
toMMR,
incompleteCarlosampling.
Not
only
do
the
i are them between $mF−1include ejection processes that
and
used
in
Fig.
13.
The
rms
of
the
relativedifferences
place.
However,thesechannels
only
in
a
3:2
while
Uranusand
Neptune
are
in
a
5:4
MMR
(for
indicatedbythelimiting
formofequation
(6);
the
blue
dashed
curve
shows
2
singlestars.
The
starting
states
have
four
giant
planetswiththecurrent(where
ℓ
∼
890
au
for
v
=
1
km
s
).
The
final
expression
rep= Noteαπℓ thatexp[relationbof(1−e)unity], the(7)
tems.where
Combining
this requirementwiththe results
our planets.
numerical
The
apparent
twovariables
observed
in values
bbetweenmensionlessconstant
orderandisplotted
includedto(asforencapsulatew
orbitsthe indicesrunothers,
throughallfourof&theof
giant
thecross-sectionsareonly
eccentricity
sections
show andnearlydependence
on a,the slope
of
the
T
T
$of our Solarw system. 448,m m (2015)
MNRAS
344–363a happenthefull form.
duringorimmediately
after
theevidence
encounter(wedenotethese
furtherof this linear
initialstate,
seebutBatyginmassesand semimajor axes
Cross-sections
arem
scaledThe
apparent
relation
betweenthevariables(as observed
indiscussion
theresulting
numericallycross-sections
the
curve
given bythetheresults)
could
thus
be
of
antwo
equipartitionsimulations,wedeterminedfind andthatcross-section
forandscatteringa Solarresentsform,
which
is applicablewhengravitational
the uncertainties
inherent
insuchthisaapproximation.
After
insertingthe eccentricfi wm
MNRAS 448, 344–363 (2015) m
cross-sections
fortheincreasing
the(bluespreadof inclination
esimulation≥limiting
0.20.
Withinmassiveplanetarysystem,small
6/5
w &theS dashed)O
curve2010;is Thepredicted
byabove analysis.
The
solid
processes
as prompt
ejections).
InLichtenberg
othercould
cases,
planets
arescatBrown
Nesvorný
2012;
Li
& red
Batyginby the factor
vb /a.
The individual cases are shown as light dotted curves,thesimulation
results)thus
beevidenceof an equipartitionlike
mixingofescape
actions(see&theLieberman
1992).
system(7)intoMorbidellia ∼13state where
the
Kozai
effect2014).
can operate
is given
bythus self−
focusing
dominates,
andimpliesalineardependenceofthecrosswhere
the
first
factorenforcesconsistency
withtheejectioncrossequation
is
per
cent.
The
cross-section
curvesareexpression
forthespeed,
we
obtainmangles
shown
in
Fig.
Theimply
figureshowsthesemimacross-sections
ities(e∼ 0.10)are easily
excitedby(seeplanet–planet
interactions;as
F planetsTand four values of velocity dispersionmmm
σ ∝the four giant
curve
Fig.
12is showsthecross-section
predictedbyequation(6) for atoisolatethedependencethecross-sectionson
teredinto !
high-eccentricity
orbits,sopaperthat theorbitscrossother.
Note
thatinthesemimajor
axisofratios
do7.not
periodratioswithwhich include
like
ofthe
the
actions
Lichtenberg& Lieberman1992).
Althoughbeyondthemixing
scope
of
this
paper,
thisissueeachshouldbe"! initial!accuracy.
""
sectionon
a. Given
thisform,Forcross-section
requiresanother
−7/5
F
vthe
aout
sections
above.
value
bthis
=
4/3,
we
a goodsimilarto thislevel
ofFurthermore,
the dependenceofb orbital
T s−1 . The heavy solid blue
1
range
of⟩casevelocity
dispersions
ofboththeAllbackgroundcluster,aconsideredresult,theAlthoughcross-sections
forincrease–obtainasthisdetermined
2for
GM
jor
axis
planet–planet
scattering
effects,
this
case
uses
With
these
configurations,in the theabsenceofofresonance,planets
exact
integervalues(althoughthey
areclose).
oflatter
the
for
the
limitingform);
cases,
characteristicof the background cluster: v b = 1, 2, 4, and 8 km∗
beyondthe eccentricity
scope
paperthepaper,
issue should
befrom√ v b =(10)exploredfurther.
≈(full210
000 au, from⟨σkozaiαπ
a 2 In1 this
+
→
απ
ℓa,
(6)
⟨σ ⟩tolength-scale.
problem,
the
orbit
speed
of
the
binary,
the
−1
−1
T
σ
0 =
1
2
fi
w
=
$
m
30
au
1
km
s
fit
the
calculated,scaledcross-sections,
as
shown
by
the
heavy
the
cross-sectionsonvelocity
dispersion
isnearlyindependent
of
,
where
the
valuesarespacedbyfactors
of
2.
The
to
16
km
s
throughournumerical
scheme–areextremely
large
and
are
not
curve
depicts
theaverage,
where
the
error
bars
depictthestandard
deviation.
willWeeventuallyexperiencecloseimportantencounters,
which in turnlead
toelements
must masses
betoput sothetheysystemin mutual
smallerplanet(byaandfactor
10),
act
more
likeα test
length-scaleℓ =chosen890
auproperly
theof
dimensionlessparameter= 7/5.
exploredfurther.
ava bpotentiallycan extract
cross-section
from
these
of(we
thedenote
encounter,
the asspeed
the from
asymptotic
speed
v ∞We
A red straight line shows the result for cross-sections with anmexponential
w FTheejectionsor collisions
thisprocessdelayedejection).
MMR,
and
this
requirement
displacestheperiodratiosarealmostinsomewhat.
semi-logarithmicplotplottedinthefigure.
where
aoutisrange
the
axis
of
the
outermostplanetof w
themasses;to cross-sections
explorea widerin evenly
a, we
also
take
thisaxis
compact
canextract
a consideredpotentially
importantcross-section
these
red
line
inhaveFig.
13.
Except
for
first
point
(eorbit=
0.1),
theofexponential
theconsider
dependence
on semimajor
starting
semimajor
a oftheresults.
The
scatteringinteractions
here
can readily
inNextwethedependence
ofspacedthecross-sections
onm
the planet.
withthevelocity
whereThewe
replaced
asymptotic
speed
vbe
∞
are
allouterroughly
comparable
planet,andthevelocity
dispersion
v btheAσ ∝−dependence
on eccentricity.
$ ∼ $w
fiby Note
mthat
m is
cross-sections
forthedelayedejectionswillconsideredlater.
None
the
less,
the
resonance
anglesofthesystem
(for
allof
three
planetsystem.
therequirement
large
mutualinclination
results.
The
scatteringinteractions
consideredhere
canreadily
inand
have
nearly
the
same
shape
as
a
function
of
!i.
These
properties
The
numerical
resultsforcross-sections
canbeunderstoodsystemtobesmallerafactorof
1.35
comparedtothatconsidcreasethespreadofinclination
anglesofbodiesinaSolar
post-encountereccentricitye(whichisequivalentto%ebecausefit(straight
red
line)
agreeswithbackgroundtheaveragevalues(solidblue
The dependence of the interaction cross-sections
on themass ofw−1
ofthecluster(orotherstellarsystem),dispersion
v
b
≈
fi
).
For
example,
the
orbit
of
Neptune
in
our
Solar
sys(1–10
km
s
wm
m
For
comparison,
wehand,also
present
theoften
results
fromaofseriesofpairs)
insection.
the theinitialstate,
asbarsrequiredWiththespreadofweinclination
anglesouterbodies in
athe
Solar
necessarybutnotsufficient
forafortheMMR.
Kozai
effectto allplayona vrole.
system.
On the crease
other
the
scattering
eventsthemselves
have
indicatethatcross-section
haspower-lawdependence
as withinfollows:
to leading
order,
expecttheplanets
as
ered
inare
thelibrating
previous
error
delineate
theuncertaintyborbits
start
with
zero
eccentricity).
For
all
four
planetsin
2 to act
curve)
toabout
3using
per
cent,
i.e.
the
difference
isspeed
much
less
the
host
star
isThe
somewhat
morecomplicatedthanfor theothervari−1/v
and haswehavedefinedthecorresponding
length-scale
ℓ
≡
GM
∗
w
F
numerical
experimentsamorecompact
orbitalarchitecture
thisinitialstate,
theSolarsystemismuch
more
compact
than
at
the
F
$
$
F
,
whereas
the
orbit
tem
an
escape
speed
of
∼5.5
km
s
b
system.
On
the
other
hand,
the
scattering
eventsthemselves
have
eccentricity.
Althoughthisbehaviourisanalogous
to
that
found
for
The
Kozai
effectthe
is ae-dependence
highlyNot
fragiletype
of interaction
itlittletest
effecton planetstighttheorbits,such as the multiplanet
to incompleteMontesampling.
only
do
the
cross(seeofSection
4).Carlo
particles,
sointhatcross-sections
should notsystems
be sensitivedue
to
−1
three
types
Solar
system,isare
similar.
Sincethebecause−1standard(whereℓ∼890
forveffect
=
1onkm
sin
).
repof
increases,
than
the
width
of
the
distributionsasmeasuredbythe
theables,
aslinearillustrated
Fig.
4.
As themassMbthe(showntheau
reddashedcurves
Fig.
1),
which
issuchmotivatedbypresent
epoch,
andthecross-sections
smaller.
WTσ the∗positively
littleplanetsinThe
tightorbits,asmultiplanet
systems.
ofa binary
the
peak
of
perioddistribution
isexpression
also
∼5
km
sthe
observedbytheKeplermission(Batalha
et finalal.
2013).
However,
thewbinaryTcross-sections,the slopemof the exponential is somewhat
involves
libration
ofin
theargumentof
periastron,
andthis star
quantity
sections
show
dependence
ona, buttheslopeof theInnearly
general,increases
infor
theinteractions
inclination
anglesarecortheatas
planetary
masses.
For
sufficiently
massiveplanets,
however,anejectioncross-sections
scale
linearly
withsemimajor
axis
a
(see
resents
thelimiting
form,
which
is
applicablewhengravitational
Nice
model
of Solarsystemformation(Gomesetal.
2005).
This trend
isthe
illustrated
insubject
Fig. 2, which
comparesthecross-sections
observed
bytheV,Keplermission(Batalha
et fit2013).
However,them of thew form mm(for furtherF
cross-sections,consideredasfunctions
eccentricity,
becomedeviations.
ofthequality
ofleadtheisefprovided
bodiesinAnother
theainouter
Solarsystemcanhaveimportantlong-term
If wetheonly
have
singlevelocitythendimensional
analysis
imsteeper. Here we consider a fitting function
can
be
toin
many
other
of precession
curveis12),
predicted
bythetheabove
analysis.
The
redsources
solid
(blue
dashed)relatedwithincreases
eccentricity.
This
resultis ofnotpower
unexpected,
increase
the measureeccentricityofone
planetcantoal.significant
perFig.those
we scalecross-sections
by
out
one
focusing
and
impliesaa lineardependencethecrossAlthoughtheinnerNice
modelhasouternumbervariations,one
feature
with
for
solar
systemswiththe dividing
standard
2intobodiesinprovided
the
Solar
systemcanof
havelong-term
ef- obtained
AThe
w
σ(for
∝
m M
fectsdominates,on differencesthebodiesthat
theyscatteredorbitsdiscussion,see
Batygin,Morbidelli&
Wedifferent
also−note stellar, important
as
given
plies
that
the
relevant
length-scale
must
beof
ℓare
=therebyGM!Ka "w! w
spacingof
thecurvesinstarting
Fig.
4(6)
the
relative
betweenthenumericallydeterminedcross∗ /V
curve
Fig.
12
shows
the
cross-section
predicted
byTsiganis
equationassteeper.
changesinbothorbitalelements
to 2011).
disruption
turbations
on
the
other
planets,
leading
to increased
vb "6/5
oforbits
a.inTheresultingscaledcross-sections
arecorrespondshown
in Fig.
13 as a of theiswiththattheGiven
giantacting
planets
could
have
formedwitha morecompact
configuration.
Tothatleadingorder,
thesmallercross-sections
obtained
fects
on
the
inner
bodiesprovided
thattheyanotherareifscatteredinto
section
on
a.
this
form,
the
cross-section
requiresw≈fimsufficiently
highinclination
angles.
Morespecifically,
the
=
⟨σ
⟩
exp
(1
− e)] ,
(11)m
⟨σ
⟩
[b
this
form
forforthecross-section
10)
involvessome0sin equation(6);inadditional
uncertainties
can thebe absorbed
into
the
for
the resonant
limiting
case(fullform);
cases,
thecharacteristic
masses)grows
e,bothsothatindividual
curvesnot self-similar.
In single
sections
used
constructing
Fig.
13day.
andexponential
form
given
initialstates.
To
illustrate
thistrend,
in theFig.(equation
8 wecases
plotaretheincreasesin
eccentricity
excitation.
By
increasing
mass
all
the planets◦of
1 km s−1
ofeccentricitye.
Inawithadditionto(shownconfiguration
than
that
ofouterthethepresent
Forthe
this
case,
we
fixof
the
forfunction
thearchitecturearedirectconsequence
with
high
inclination
angles.
Morespecifically,
if thelength-scale.
In angles
thisproblem,
orbit
speed
of
the
binary,
the
2,
then
the
inclinationof sufficiently
theorbitsarelargerthanw theF of the smallermauFT
1 out primarily
extrapolation:
numerical
simulations
werecarrieddimensionlessparameterα.
Finally,
we
note
that
for
a39.
velocity
dis-cent.
ℓ=
890
au
and
thethedimensionlessparameterα=7/5.
◦ length-scalethespreadofinclinationangle!iversusthechangeineccentricity
to
that
of
Jupiter,theresultingsolar
systemsaremoreexcitable.
The
−
spite
of
this
complication,
we
can
still
fit
the
cross-sections
withbyequation
(7);
the
rms
of
these
relative
errors
is
∼12
per
2,
then
the
inclination
anglesoftheouterorbitsarelarger
than
39.
2
as
the
light
dottedcurves),theaverage
is
shown
as
the
heavy
blue
Kozai effectoperate1962; Lidov1962),and theof the(Kozai
encounter,the orbit
speed
of innertheasymptotic
speedcanv∞
w obtain a good fit M
mau a − = w
T versus of
Figure 17. Scaledfor
cross-sections
eccentricity
increase!esystem.
(equiv- None the less, the−1w ⟨σ ⟩0 = 1000
σ ∝where weforthethearchitecture
thethecurrent
Solar
sKozai, thegravitational
focusing
term
dominates
persion vlargest
Nextwe(equivalently,
considerdependenceofcross-sectionsontheb = 1 kmthepost-encountereccentricity
since!e=(equive).
The of theincreaseineffectthecross-sections,which
occursfor
Jupiterandoperate(Kozai1962;
Lidov1962),
and
the
inner
Figure17.
Scaledcross-sections
versuseccentricity
increase!ealently,
post-encountereccentricity
e)
for
solar
systems
interacting
withapower-lawfunctionofstellarmass,
althoughtheaccuracy
Next
weconsidertheeffectsofthevelocity
dispersionoftheportionsof velocity
the
Solar
systemcan vbeexcited
over
the
age
of
the
syscurve,
where
the
error
bars
depictthestandard
deviation.
Thislatterareallroughly
comparable
planet,andthedispersion
m
w
m
b
T =
fi =
w
F
T
fi
outermost planet
isMNRAS
alwaysthemostaffectedby fly-by interactions,448,
344–363(2015)
eccentricity(whichis equivalent
tobut%ebecauseby a factor
oflow
30.eccentricities,
variables
areehave
in
fact
well
correlated,
the
of
is
only
aNeptunefactormostcross-sections
alently,eccentricity
e)
for
solar
systems
interacting
withportions
of
the
Solar
canof
be∼2;
excitedover
the age ofpost-encounterthesyssingle
stars.twoThestartingapost-encounter
statesfour
giant
planetswiththecurrent
Combining
thisrequirement
withtheresultsofourquantity
providesmeasurethe
spreadinto
thevaluesof rangethecross).cluster
For example,
the
orbit
ofsystem
in
our
Solar
sys(1–10tems.
kmfor
s−1
w
wlinearlym
M
FSmapproximation
is of
not
expected
be
as
high
as
in
thepossible
previous
backgroundenvironment.
As
shown
in
Fig.
3numericalintheprevious
Tm w wmM
and axes
thecross-sections
scale
with
semimajor
axis
toa good cases.mtheorbitswithzero
eccentricity).
For
all
four
planetsinall
Thesimulations,limiting
formofequationis Theselinearinthestarting
singlestars.
states
have
four
giant
planetswiththecurrent
14.
Scaledcross-sections
versuseccentricity
increase(equivamasses
and start
semimajor
our startingSolarsystem.
Cross-sections
arescaledtems.
Combining
thiswiththe semimaresultsour(equivanumerical!iFigurevaluesfor ofaThegiven
eccentricitye=!eislarge.
Asa fromresult,the mexperiencesmallerchanges.
results
areorbitgenerally
consistentwe
findthat∼5.5the(6)
cross-sectionforscattering
aspeedSolarofFigure13.
Scaledcross-sections
versuseccentricity
increase!e−µ!e inm M−,thewhereasthetem hasanescapespeedofkm
s−1requirement
sectionoverthevariouscases.
The
standard
deviationvaries
m
m
T
w
m
m
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∝
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approximation.
,
and
vary
the
inWe
thus
considerascaling
of
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form
⟨σ
⟩
∝
M
section,thecross-sections
vary
with
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andsemimajor
axes
our Solarsystem.
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are scaled
three
types
of
Solar
system,thee-dependence
similar.
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different
byaeachthefactorfigure
vblently,/a.weThe
individualcases
are ofshown
light
dottedcurves,simulations,
we
find
that
cross-section
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jor axis
awith
oflently,
theTo oursee
howwell
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we plansystem
apeak
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where
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is no
given
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in
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averaged
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theplanet.
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that
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i.e.
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per
cent
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m eccentricitymw !e (equivaNext
weconsiderthewithscalingbehaviourofcross-sections
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the
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andvaluesofvelocity
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µthe
to
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−
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−7/5
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authe
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⟩kozai
Fig.
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by
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one
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shown
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light
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depicts
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15
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b .Fig.
choiceofcross-sections
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210 000,⟨σthe⟩kozaimasses. The
have
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au absorbed
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/aare
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additional
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semimajor
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athe
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deeccentricity.
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the
heavy
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MNRAS
448,
344–363(2015)
dependence
on theeccentricity.
−2−1 sufficientsystem.
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the
requirementoflarge
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m
m
w
m
m
w
T
m
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necessary
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m four planets
w for four choices of stellar mass
1 km s ,We
the havegravitational
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persionv b = order).
shoulddisplayonthe
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velocity(toleading
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theindividual
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the average,
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standard
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s to
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m
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F
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The Kozai
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becauseit to playeccentricity.
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andthenchoosethenearest
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H
by
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velocity
dependence
for these
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w
and
find
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the curves
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⟨σ
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and
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nearlystraight
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depicts theb .values
quantity
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compact
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moderately
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mw
average, where the error
The red straightlogarithmicplot,
so (6)
thatthecross-sections
on overforThe
γ can
≈
7/5.
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thislibration
choicepower-lawindex,
weand
plot
the
corresponding
standard
deviation,
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ism
largerthevariousstandard
deviationvariesfromthebinary cases.
cross-sections,
theof the exponentialissomewhat
involves
ofofthedependence
argument
ofof
periastron,
thissection
quantity
steeper.
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fittingfunctionform
subject
to manyothersourcesofprecession(for furtherw
mThemofslopewthe
σ which∝ thanM −is for themmS m=
jor axis
a beofeccentricity
theplanet.
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expression
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we≈
6/5,
earlierfortheoptimal
value
of
the
index
γ
=
s
line
shows the result for cross-sections with a purely exponential dependence
nearlyexponential.
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purposesof noteillustration,
we17 percases
steeper.
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we consider
fittingfunctionofFig.
can isbesubject
to manyothersourcesofalsoprecession(forfurtherabout
cent
ofconsideredthecross-section
ata low
eccentricity
ethe
=form
0.10with Figs
scaledcross-sections
versuspost-encountereccentricity
in
Fig.
14,
previously(compare
15
13
and
14).
discussion,see
Batygin,Morbidelli&
Tsiganis
We
"
!
!
"
fi
w
m
A
m
m
m
m
6/5
m
w
plot the ejection
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versussemimajor
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v one thanabars depict
somewhatsmallerthe value
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error
standard
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heavy straightγredline on eccentricity. m ∼Fuseformanexponential
fittingfunctionof theform
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Batygin,Morbidelli&
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10)
involves
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12.
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givenequation(7)
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7/5
thenumerical
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M Theaurmsw1errorw⟩esemimajor
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s
/a).
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inthefigureshows
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equation
7).
betweentheexponential
line
and
the
function
of ⟨σ
the
axis.
This
trend
holds
for
solar
systems
reducedcross-sections
are
shown
in
Fig.
17.
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light
dottedcurves2
and aused
rms auof thedifferences between power-law form,m∼ though
T fithe results depart somewhat more
extrapolation:
simulations
werecarried= απℓofexp[(1the− numericale)], system.
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bs = 8/5.
wheregoodinfitFig.
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None theless,
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m
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2values of v (which∼ mT
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show
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ared
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⟨σparameter,⟩inbper cent.
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is
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the (star
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ane ary−1−2lineCross-sectionsforplanetary
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357
).The
The
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lie in
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compact
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theredstraight
inthe
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fly-byinteractions,w==fi
specifiedin thissection,we
generally
searchof10, curvefindthefirst averagefactorenforcesconsistency
withtheejectioncrossequation
(7)
is
∼13
per
cent.
cross-section
curvesarethus
self- with a w
rms difference
betweentheexpressionofequation
(11)
and
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shows
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all
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thecross-sections
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scale
linearly
withsemimajor
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abygood
−
m
F
m
F
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forincreasing
thespreadofinclination
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nar
e
w
σ
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difference
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the ofoverthethisentirecollection
and
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also
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results
ultraandthecross-sections
scale
linearly
semimajor
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toa goodthat givesdetermined,theminimumrms
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round
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H&mRm
sections
considered
above.
the
value
bwith
= for
4/3,
we obtain
goodsimilartolevel
Furthermore,the(11)
dependence
approximation.
fiof accuracy.
w ∝ −overconsidered
a
range
of
velocity
dispersions,also
show a nearlynumericallydetermined,
scaledcross-sectionsforsingle
stars
is
W
w
σ
deviations.
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error
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error
approximation.
number(ratioofrelatively
small
integers).
compact
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order
The
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betweenchangesinthevariables
sin
!i
and
e
only ∼8 per cent.
Tmw
Nextfitwethe scaling
behaviourof the cross-sections
toconsider
the calculated,scaledcross-sections,as shownforby the heavy
the cross-sections
on∼velocity
dispersion
is nearly independent of
T
m
T
fi
σ
∝
‹
only
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weconsiderthe first
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forwe canthe
ofdependence∼15
percross-sections
cent,
isfor
comparable
to, but
largerNow
comparesingle starsaxis
withprovidesintriguing
topic
forstars.
additionalwork.
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leadingorder,
interactions
withpassingsingleAs forthe (ecaseofm indicates thatww the crossredanline
in Next
Fig. 13.
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theexponential
on whichstarting
semimajor
athoseofsomewhat
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448,(2015) m w
mwfor
wwith344–363
Nowwemcanthe cross-sections
single stars
thoseinteractions
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singlestars.linearly
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that
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binaries. TheThecomparison
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scalings
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systems,
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terms
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cross-sections
on scalings
the mass of section should
O
w
=
=
fi
for
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comparison
iscomplicatedbythedifferent
−η
MNRAS
448,
344–363(2015)
systems,
we
expectthecross-sections
toscalenearlylinearly
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−1
m the
The
scaleddispersion
cross-sections
shown
in Fig.
17 for single-star interac- ⟨σ ⟩ ∝T v . Over the range v b = 1–16 kmws A
of the two cases withvelocity
and thedifferent
exponentheforms
semimajor
axis
a ofabouta givenplanet.
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curve) tothe
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the
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is somewhat
morecomplicatedfor theother varib
H two
&
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a givenplanet. In
the$tialnearlylaws for theeccentricity
dependence.
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fix dependence
ideas,! # m howwmthism mscaling law works,w η ≈ 7/5. !To illustrateshow
a Rnearly
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on the post-encounter−equal1spacing onthesemimajor
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11 indicates
thataddition,
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"
fit
occurs
for
well
of
the
star
increases,
thanthe equal
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ables,
asillustrated
in
Fig.
4.
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the
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M
M
∼
m
∗
T
m
w
−1
tial
laws
for
the
eccentricity
dependence.
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ideas,
consider22
spacing
on thelogarithmic
plotonoftheFig.
11 indicates
thatbenchmark
thecase where the velocity dispersion v b = 1 km s
and display
Z ∝ power-law
sin(i/2).
(9) the"cross-sections∝ eshoulddependence
velocity
7/5
σ=
we plot =the scaledtheconsidered
functions
Another −γ
measureof thequality dependenceof the fit isonprovided
m − cross-sectionsTm of eccentricity,
km becomes−1
thecross-sections,benchmark
case
where
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dispersionv b = 1w
2deviations.
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spower-lawtheforvelocity
the backgroundcluster. Here,thecross-sections
for binary-star. The display
velocity−γ
dependence
for these
dispersion,
suchcross-sections
that ⟨σ ⟩ ∝ vb should−sforthe backgroundHere,
the single
cross-sections
binary-starsteeper.
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4 (forstellar in Fig. ∼the relative
differences
betweennumericallydeterminedmw the figure
m to them
16. Each
curveincorresponds. The
velocity
dependence
for these
dispersion,that⟨σ ⟩two∝thevbvariables% MNRAS&The
apparent
relation
betweenthe(as observedin crossinteractions
are∼4.2
timesspacing
largercluster.
than
those
for
stars
at fordifferent
single-starcross-sections
issuchmoderately
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than
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found
"−
! light dottedinteractions
are
∼4.2
times
larger
than
those
for
singlestars
atw of velocity dispersion.
masses)grows withthat
the curves
are not self-similar. In result of one
sections
used
constructing
Fig.
13
and
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form
given
single-starcross-sections
isindexmoderately
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those
found
The heavy blue curvethehigh end of
the eccentricity
rangee,eso= 1.
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M
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"
!
the
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the
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m
≈
6/5,
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γ
spite
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this
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we
can
still
fit
the
cross-sections
withbyequation
(7);
the
rms
of
these
relative
errors
is
∼12
per
cent.
s
Mof the eccentricsomewhat
the valuebinary cross-sectionsγ ≈ 7/5.1992). cross-sections are ∼3.2 times larger at the low end
shows the average of the scaled cross-sections,likemixingsmallerof thethan
actions(seeforLichtenberg& Lieberman
w where them errorσ 0.10.
= Averaged
are ∼3.2overat thelow end theof theeccentricsomewhat
smallerthan
the valuebinarycross-sections
γity≈the7/5. where
m mm
range
e=
thelargerspan
a cross-sections
power-lawfunction
oftimesstellarmass,
althoughaccuracyof the bars depict
Next
we
considerthethiseffectsoftheforvelocity
After scalingoutthesemimajor
axis
andvelocity
dispersion,
the be ofM⊙of eccentricity
Althoughbeyondthescope ofpaper
paper,
thisissuedispersion
shouldthe standard deviations. The mean
size of the error bars
itytherange
where
e = 0.10. Averagedoverthespan of eccentricity
Aftercluster
scalingout intheFig.
semimajor
axis dottedandFig.
velocity
dispersion,
the here,
T mmw
consideredbinarycross-sections
aretolarger
bya asfactorm
reducedcross-sections
are
shown
17.
The
light
curvesapproximationisnotexpected
be
as
high
in
the
previous
cases.
backgroundenvironment.
As
shown
in
3
in
the
previous%&
explored further.
+ differences−f σ
σ = f to"− cross-sections are larger! the binary
correspondsof ∼6 permcent,−−considered
here,
by a factor
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G. Li and F. C. Adams

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354

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López,1 J. A. Acosta-Pulido,2,3 R. Estalella,1 G. Gómez4,2
d B. Garcı́a-Lorenzo2,3

artament d’Astronomia i Meteorologia (IEEC-UB), Institut de Cièncias del Cosmos, U. de Barcelona, Martı́ i Franquès 1, E-08028 Barcelona, Spainuto de Astrofı́sica de Canarias, E-38200 La Laguna, Spain
artamento de Astrofı́sica, Universidad de La Laguna, E-38205, Tenerife, Spain
28. “3D S.A.
kinematics
ofBreñathe Baja,near-IR
HH223
outflow in L723 R.”!
Project Office, GRANTECAN(CALP), E-38712
La Palma,
Spain.

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ABSTRACTIn this work, we derive the full 3D kinematics of the near-infrared outflow HH 223, locatedin the dark cloud Lynds 723 (L723), where a well-defined quadrupolar CO outflow is found.
HH 223 appears projected on to the two lobes of the east–west CO outflow. The radio continuum
source VLA 2, towards the centre of the CO outflow, harbours a multiple system of low-mass
young stellar objects. One of the components has been proposed to be the exciting source ofthe east–west CO outflow. From the analysis of the kinematics, we get further evidence on therelationship between the near-infrared and CO outflows and on the location of their exciting
source. The proper motions were derived using multi-epoch, narrow-band H2 (2.122 µm line)
images. Radial velocities were derived from the 2.122 µm line of the spectra. Because of theextended (∼5 arcmin), S-shaped morphology of the target, the spectra were obtained with themulti-object-spectroscopy (MOS) observing mode using the instrument Long-Slit Intermediate
Resolution Infrared Spectrograph (LIRIS) at the 4.2 m William Herschel Telescope. To ourknowledge, this work is the first time that MOS observing mode has been successfully usedin the near-infrared range for an extended target.
Key words: ISM: individual objects: HH 223 – ISM: individual objects: LDN 723 – ISM: jets
and outflows.

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on February 12, 2015

R. Lopez, J. A. Acosta-Pulido, R. Estalella, G. Gomez, and B. Garcia-Lorenzo (Spain)!
http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stu2607!
pted 2014 December 4. Received 2014 December 1; in original form 2014 November 12!
2015, MNRAS 447, 2588-2602!

Figure 3. J-band spectra of the slitlets covering the knots with detected
emission lines of the HH 223 H2 outflow, obtained by averaging the signal
within the corresponding slitlet. Intensities have been scaled relative to the[Fe II] 1.257 µm line intensity peak of the HH223-A knot.

!

tions at scales smaller than the length of the slitlets, we extracted,
in addition, the one-dimensional spectra of each of the H2 features
identified along the HH 223 outflow. The kinematics of HH 223 wasderived from the spectra. The spatial brightness distribution of neutral and ionized gas outflow was also explored from the 2-D MOS
spectra (see Fig. 6).

ページ 6/10

Figure 4. Same as Fig. 3, but for the H-band spectra. Intensities have beenscaled relative to the H2 1.75 µm line intensity peak of the H2-NW2-d knot.

MNRAS 447, 2588–2602 (2015)

cs of the outflow

cience frames usjacent sky frames.

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observµm (top) and of theed within the slitlets
that corresponds to⃝
C 2015 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society

3.1 Emission brightness distribution
3.1.1 The neutral outflow

Figure 5. Same as Fig. 3, but for the K-band spectra. Intensities have beenscaled relative to the 2.122 µm line intensity peak of the H2-NW2-d knot.

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on
Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on February 12, 2015

The protostellar systemum)の輝線をWilliam
embedded in SMA 2 seems toHerschelbe assoline 2.122 um (他 1.7umや[FII]1.26
N T RO D U C T I近赤外にあるH2ON
ciated with the large-scale outflows detected in other wavelength
2592
R. López et al.
solated dark cloud
Lynds 723 (L723),
locatedat a anddistanceof
ranges,as proposed from previous deep narrow-band images of theTelescope
4.2m,
MOS
LIRIS用いて、L723にあるoutflowを観測した。!
± 150 pc (Goldsmith et al. 1984) is one of the few sites where
L723 field. In the optical range, it drives the Herbig–Haro (HH) obH2 lineでは複数のHH天体
ll-defined quadrupolar
CO outflow has been reported(HH233)が見られ、shockで励起されたガスのknotsがある。!
(twoject 223 of the Reipurth Catalogue (Reipurth, 1999) first detected byate pairs of red-blue lobes; Lee et al. 2002, and references
Vrba et al. (1986) as a ‘linear emission feature’, and later resolvedin). The 3.6 cm観測ではLine分光で視線速度を測定した。また複数期間での観測をし固有運動を調べた。!
radio continuum source VLA 2 (Anglada,
into several knots by López et al. (2006). The spectra of these knots
ı́guez & Torrelles 1996), located towards the centre of theare characteristic of shock-excited gas, as revealed from long-slit
outflowが知られており、H2
utflow, harboursCO
the low-masssource that powers the outflow. lineでの結果と比較。!
spectroscopy (López et al. 2009). The emission of the knots show
sco-González ettitleの3D
al. (2008)kinematics
and Girart,
Rao&
Estalella(2009)
complexpattern, bothin itskinematics and physicalconditions,は including proper motions and aradial
velocitiesでIR
H2 line(2.122um)のHH233
knotsの駆動源を同定した。!
t that VLA 2 is a L723;
multiplesystem
of
four
(VLA
2A,
2B,
2C
and
as
derivedfromIntegral
Field
Spectroscopy(IFS)
observations
by!
young stellar objects(YSOs),
andLópez et al. (2012). In the near-infrared range, Palacios & Eiroa
distance300±150
pc!that one of the components
COthe
outflow(2 separateof red-blue
A 2A) is powering
east–west
CO pairs
outflow.
Later lobes)!
on, Girart(1999) detect in a K-band image several H2 emission nebulae, lo中心ではVLAの観測で VLA2が4つのYSOs(VLA 2A, 2B, 2C, 2D)があることが知られている。VLA 2Aが東西のCO outflowの駆動源である。!
cated at both sides of VLA 2. Later on, López et al. (2010) obtain(2009) detect dust emission at 1.35 mm, resolved into two
SMAの観測1.35mmで二つの成分が知られている(SMA 1 and SMA 2)!
deep images of the L723 field through narrow-band filters cenonents (SMA 1 andSMA 2). SMA 2, in a more evolved stage,!
tred on the [Fe II] 1.644 µm and H2 2.122 µm lines. The images
bouring the multiple
low-mass protostellar system VLA 2. Inshow a set of H2 emission features, distributed from the south-east
ion, they report emission from theLIRISSiO MOS
5–4 line
towardsof HH
223 the2591
2592
R. López et al.
↓Figure3-5,[FII]1.257um,
1.75um,2.122umthe north-westof the L723 field, extending along ∼5 arcmin
sources, which shows
an elongatedmorphologythat follows H2 to(∼0.5 pc for a distance of 300 pc). The H2 emission features areirection of the east–west CO outflow near the exciting source
found projected on to the lobes of the east–west CO outflow, withell as that of the near-infrared and optical outflows reported
an S-shaped morphology, and are proposed to form part of an H2
23.
outflow, which also has optical counterpart at the regions with low
visual extinction, given rise to the HH object 223. In contrast, the[Fe II] 1.644 µm image only shows emission associated with theail: rosario@am.ub.es

2592

Figure 4. Sa
scaled relativ↓P-V diagram;→H2 2.122 umのL723領域のimage.
2590

LIRIS MOS of HH 223

長方形はMOSでの分光のスリット。円形(eg, J22)はmaskしたところ。

2593

R. López et al.

SMA 2はK1,K2付近にいる。

!

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Slitlet

Knot

Offset b(arcsec)

VLSR c(km s−1 )

s1
s2
s3
s4
s5
s6
s7
s8
s9
s11
s12
s13
s14
s15
s16

N1
A
B
C
D-E
F
N2
K2
K1
H2-NWabH2-NW3H2-NW2ab
H2-NW2d
H2-NW2eH2-NW2fg

−83.9
−67.0
−64.0
−59.9
−55.4
−51.4
−37.7
−8.3
−3.9
40.3
82.8
101.9
112.8
118.2
123.9

−72.1 ± 12.4
−26.3 ± 9.7−21.3 ± 8.2−7.0 ± 6.2−33.2 ± 12.4
−39.2 ± 14.0
−30.3 ± 6.5−24.0 ± 9.8+7.3 ± 8.0+1.5 ± 8.3+43.3 ± 11.5
+49.0 ± 10.6
+36.0 ± 6.1+39.9 ± 12.6
+25.6 ± 7.9

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on February 12, 2015

Table 3. Radial velocities (VLSR )a of the H2 2.122 µm emission within the full slitlet aperture.

Notes. a From Gaussian fits to the H2 2.122 µm line of thespectra, averaged for the full slitlet apertures (see the text).
b Offsets from the position of SMA 2 to the centre of the
Figure 6. Position-velocity maps of the H 2.122 µm line through the slitlet labelled in each panel (black contours). The PV maps of the [Fe ] 1.644 µm aresuperposed (red contours) for the slitlets where [Fe ] emission was detected. The PV maps of the H 1.748 µm line are also displayed (green contours) for
slitlet.
Typical values of the error are ≤ 0.5 arcsec.
these slitlets to check the reliability of the spatial displacement between the H and [Fe ] emissions. The knots intersected by each slitlet have been labelledc The error values correspond to the error of the Gaussian fit
to help with the identification.
added in quadrature with the rms residual of the wavelength(dome flats, hereafter) spectra were used to determine2600
the correc-R. López et al.
calibration.

Figure 1. Image of the L723 field obtained with LIRIS through the H2 2.122 µm line filter (the continuum has been subtracted). The rectangles represent theposition of the slitlets of the mask designed for the multi-object spectroscopy (MOS) observations. The knots of HH 223 enclosed in the corresponding slitletof the mask have been labelled according to the nomenclature of López et al. (2010). An additional slitlet, labelled J23 was included to check the individual
frame quality. The four circular appertures, labelled J10, J14, J22 and FJ1, used for mask centring, are also drawn. North is up and east is to the left.

!

Table 2. 2596
Configuration
the mask.et al.
R.ofLópez

II

2

II

2

2

II

López et al. 2
each slitlet. T
each of these
slitlet encomp
1 arcsec wide
the velocity ofas the offsetof the knot, isthe knots proj
outflow are plwill be usednear-infraredcomplex kinem
which confirm
HH 223 optica
additional H2in the previous
Finally, andpattern foundof the knots fr
in most of th
and 10 (green
good agreeme
Concerning thwith those derbrightness disµm line are als
the peaks of th
general dtrend fVTthe(km1.748s−1 ) µm∼40–60 km s−
25.3
± SNR
2.0 lev
same

SlitletLengthCenter Position→CO
outﬂowαID (arcsec)

2000

δ 2000

PA
(deg)

!

・H2 lineによる大局的に視線速度は双曲流パタン(bipolar pattern)を示した。SMA2に対して、南東方向にnegative velocity(blue shifted)
で北西方向にpositive (redshifted)。この結果

は、CO lobeとよく一致し、H2とCO outflow
は同じ駆動源であることを強く支持する!

・固有運動は他の低質量原始星で知られる典型的なH2 outflowの値の範囲内であった。

blueshifted と redshiftedのH2 knotsの固有運動は同程度の値である。一般的な傾向として、↓固有運動

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on February 12, 2015

MNRAS 447, 2588–2602 (2015)tion. Next, we used the a-priori mask design information to trace
Table 6. Positions and proper motions of H2 knots.
the slitlet positions and their limits from a distortion-free dome–flatframe. Then the 2D spectra from each slitlet were extracted followKnot
xa
ya
µx b
µy b
ϵxc
ϵyc
ing the limits determined previously and were calibrated indepenN1
s1
6.0
19 17 59.034 +19 11 42.10−95
HH 223(arcsec)(arcsec)(arcsec yr−1 )(arcsec yr−1 )(arcsec)(arcsec)
dently.
The
wavelength
calibration
wasdetermined
from
argon
and
A
s2
3.0
19 17 57.970 +19 11 50.02−70
xenon lamp exposures obtained with the same instrument configuA−60.6−29.8 −0.0177 ± 0.0014 −0.0015 ± 0.0015
0.00590.0057B
s3
2.5
19 17 57.717 +19 11 49.51−70
C-North−54.2−27.0 −0.0080 ± 0.0003 −0.0029 ± 0.0006
0.00120.0027 12.2 ± 0.5
1
ration through the MOS mask. The arc spectra for each slitlet wereC
s4
3.5
19 17 57.501 +19 11 52.71−70
C-South
−52.4−29.2−0.0126±0.00130.0005±0.00200.00530.009717.9 ± 1.8
extracted in the way described before. The initial wavelength caliD-E
s5
4.5
19 17 57.087 +19 11 51.33
−90
E
−47.3
−28.2
−0.0094±0.0018−0.0015±0.00050.00810.0022
13.5
±
2.5
F
s6
4.5
19 17 56.697 +19 11 50.04−70
bration was determined as in the ‘long-slit case’ following the usual
3.2.2 Kinema
F0−46.4−31.6 −0.0076 ± 0.0003 −0.0057 ± 0.0040
0.00140.0192 13.5 ± 3.5
12
N2
s7
13.0
19 17 55.660 +19 11 53.80−70
tasks in IRAF (i.e. combining identify, reidentify and fitcoords). The
F1−43.0−29.9 −0.0122 ± 0.0026 −0.0005 ± 0.0057
0.01140.0261 17.4
3.7
We ±derived
th9
K2
s8
2.0
19 17 54.100 +19 12 13.60−70
accuracy of the wavelength calibration was then checked using a set
K2−5.5−5.9 −0.0076 ± 0.0024 −0.0110 ± 0.0007
0.01000.0036 19.0
± 2.1
1
locity
from
th
K1
s9
5.0
19 17 53.950 +19 12 18.00−87
of bright, isolated OH sky lines present in the observed wavelength
K1-West−3.4−1.6 −0.0008 ± 0.0004 −0.0075 ± 0.0015
0.00200.0071 10.7 ± 2.1
11.257 µm andstar J23
s10
4.0
19 17 53.170 +19 12 33.20−75
H2-NWc31.7
16.5
0.0063 ± 0.0014 −0.0003 ± 0.0001
0.00620.00029.0 ± 1.9−
ranges. It was noted that wavelength offsets were present, which is
given in Table
H2-NWabs11
15.0
19 17 51.235 +19 12 40.67−40
H2-NWb
34.5
20.3
0.0131 ± 0.0008
0.0026 ± 0.0005
0.00320.0023 19.1 ± 1.1−
expecteddue
tospectrographflexures.
The
determinationofradial
H2-NW3s12
5.0
19 17 48.440 +19 12 57.50−70valuesappearH2-NWa36.3
22.5
0.0078 ± 0.0001 −0.0005 ± 0.0003
0.00010.0012 11.1
± 0.1−
velocities from emission lines is a crucial aspect of this work, thereH2-NW2abc
s13
16.7
19 17 47.560 +19 13 14.40
−70
These
velociti
H2-NW374.7
37.7
0.0098 ± 0.0031
0.0061 ± 0.0036
0.01400.0171 16.4
± 4.6
−5
fore we needed to refine the wavelength calibration by applying an
H2-NW2ds14
4.5
19 17 46.643 +19 13 13.81−70eby[Fe II]−6H2-NW2b1
86.7
52.9
0.0028 ± 0.0021
0.0011 ± 0.0016
0.00890.0076
4.3the± 2.8
offset determined from the wavelength difference observed in the
H2-NW2es15
4.0
19 17 46.101 +19 13 10.85−70
H2-NW2c
94.7
56.3
0.0059 ± 0.0006
0.0101 ± 0.0002
0.00280.0008 16.7
± 0.5−
(tracedby theFigure 7. Radial velocities obtained from the line centroid of a Gaussian
mentioned OH sky lines. After this correction was applied, the acH2-NW2fg
s16
8.0
19 17 45.771 +19 13 14.40 −110
H2-NW2d98.0
54.1
0.0018 ± 0.0007
0.0141 ± 0.0014
0.00280.0056 20.2 ± 2.0−
fits to the H2 2.122 µm line of the spectra obtained by averaging the signal
curacy reached for the wavelength calibration was better than 0.35
H2-NW2e107.5
51.2
0.0173 ± 0.0023
0.0145 ± 0.0011
0.01040.0048 32.1 ± 2.7−
Figure 8. H2 image of the L723 field (grey-scale) superposed to the contours of the redshifted and blueshifted CO outflows from Lee−1
et al. (2002). The red
within the full aperture of the corresponding slitlet. Offsets are relative to
107.5
56.2
0.0068 ± 0.0007
0.0031 ± 0.0019
0.00320.0074 10.7
−6
for theHH2-NW2f
Å in all the observed wavelength
range (i.e. 4.9 bykmthes crosses.
2scale of
3.2.3± 1.5
Individu
and blue open circles of different sizes represent the radial velocity obtained from the 2.122 µm line at −1
the knot positions marked(ThethepositionofSMA
2.
2.122 µm, and 6.3 km s and 8.3 km s−1 for the 1.644 µm andexposuresof 600s to avoid
the bright sky lines andNotes. a Offsets from the position of SMA 2, α(J 2000) = 19h 17m 53.s 694, δ(J2000) = 19◦ 12′ 19.′′ 68, of the intensity peak in t
velocityis shown
at thesaturation
bottom rightbycorner).
In the followi
1.257 µm [Fe II] emission lines).
to compensate for their time variability. The observations wereimage. For the more extended knots, Cs and H2NW2f+g, the position of the centre of the box used for the proper motion determ
is worth andnotingthat theHH 223b Error
H2 outflow
lies
A check of the reliability of the wavelength calibration in theis given. The x-axis pointsItwestwards,
the y-axis
northwards.
values are√the projected
error in the properfoundmotionatfitsmal
to thperformed repeating the sequence Object-Sky (OSOS). The sky
on to thepairlobes ofoneproperof thetwofit,bipolar
MOS
thr
the Hαand close[S II] linesFig.
10 displays
close-upof Fig.
more detail thewavelength
long-slit
through
HH 223-A
show
epochsobserved. c rms residualof theknotofpositionsin themotionϵ = σ CO1 − outflows
r 2 , where σ is thewith
standard
deviatrange
to thein line
usedspectra
to derive
the kinematics
exposures
were
obtainedain
an empty
field,9 showing
offset bywith∼1 arcmin,d Proper(thelarger,east–westone)
detected
in
L723.
As
a
generaltrend,
HH
223-A
t
line:
a
changeinvelocbehaviourofthevelocityasafunctionoftheangulardistance
to
the
same
behaviour
as
in
the
case
of
the
H
r
is
the
correlation
coefficient.
motionvelocity,
assumingadistance
of
300
pc.
2
(H2 2.122 µm) can be seen in Fig. 2, which shows the line profiles
roughly perpendicularly to the outflow axis, since the observed field−1
the H2 knots with blueshifted velocities are projected on to theFive slitlets
betweentwo condensationsof HH 223-A,
ity of(top∼60
km s and
SMA
Thetargets
spectrumof each knot
averaging theof this line
panels)of the theseOH 2.073µm sky line (bottom
is crowded
and2.ourare extended
and wascloseobtained
to each byother.
blueshifted CO outflow lobe, while the H2 knots with redshifted
ering the knot
easterncondensation(HH the223-A1)
mosttheblueshifted.
signalwithin theaperturesquotedTable
and thecomplex
velocities werepanels) beingof the thespectraobtainedby averaging
signalthe
withinfull
The data
reduction
processof thisdatainset
was4,rather
velocities are projected on to the redshifted CO outflow lobe, as∼22 arcsec in
1.644µm [Fe II]apertureFurthermore,
velocity
trendofshown
by the 2.122µm lineof
throughderivedLIRISDR
from the1.748µm andprocedures
2.122 µm H
2 , andof the slitletthe
labelled
in eachthe panels.
The position/IRAF
standardwith
more
dediand combinedcanbeeasilyvisualized
in
Fig.
8.
This
spatialcoincidence
betweenthat this struct
)
and
the
ionized
lines
(e.g.
the
2.224
µm
and
lines.
Bothemissions,theneutral(traced
by
H
HH
223-A
is
also
found
in
other
H
2
the corresponding rest wavelength has been2marked with the dashedcated correction steps developed in IDL. For the standard
procethe CO and H2 velocity signs gives support to the existence of asion feature’ rII]) appear blueshifted. As expected, the velocities
substructures
found(traced2.406
theappears
two H2 shiftedverticaltheline.
Noteµm).
thatInthecontrast,
emissionfrom therestwithin
dures, weused bythe[Feroutines
available for the LIRIS MOS mode
physical relationship between the H2 and CO outflows.
Herbig–Haro oII] are more blueshifted than those derived for bothderivedfrom [FeTheHH 223-A
were
not case
identified
PV map of the [Fe II] 1.644
wavelength
positionin theof theinHthe
package.
first step was to determine and correct
in the LIRISDR
2 2.122 µm line, while it
Furthermore, we searched for changes in velocity at spatial scales
Fig. 6 (leftwhich in turnaretheconsistent
betweenthem.
II] emission peak coincides with
H2 lines,distortions
line. atTheof the [Fefor
appearsµm
centredtheposition
rest wavelength
the sky line, as should
the geometricalalongspatial axis
in orderto Accordingly
have
sion
lines at 2
smallerthanthelengthoftheslitlets.
In
Fig.
6(blackcontours),
wecounterpart
is foundfor the otherwith what was found in the PV maps (see Fig. 6) and in the H- andbe expected.
HH 223-A1,
while
no clear [FeWe thenconcluded
thatII]thewavelength
calibration
the spectra aligned within the CCD rows. Dome white lamp
2.122µm
emission
1.644µm (redplot
position-velocity
(PV)
maps
of
the
H
2
H2 substructure (HH 223-A2). Hence, we confirm that there are two
K-narrow-band images (López et al. 2010), there is some shift bethrough all the slitlets. From these maps, and comparing with theby the slitlets,
unresolved clumps within HH 223-A, with different excitation contween the spatial distribution of the [Fe II] and H2 emissions: offset
uum substract
H2 K narrow-band continuum-subtracted image of the field (see≥ 1 arcsec(2015)
are found between the peaks of the [Fe II] andditions, which may be originated in shocks with different strengths.
MNRASvalues
447, 2588–2602
H2 emissions in knots HH 223-B, -C (that corresponds to -C NorthIn spite of the lack of detection of H2 emission from the knotin H2 ) and -D. Furthermore, there are some substructures that were
HH 223-B in the narrow-band images, we were able to detect H2emission through the slitlet positioned on this knot (S3). From thedetected in H2 but did not appear in [Fe II] (e.g. HH 223-A2) whilesome [Fe II] substructures were not detected in H2 (e.g. a third conH2 2.122 µm line, we derived the velocity at the intensity peak of thedensation lying between HH 223-F0 and -F1). As it was noted, theseemission, and at a secondary weaker substructure lying to its southfacts may be indicative of shocks of different degree of excitationeast (named HH 223-B-peak and -B-east, respectively; see Table 4).
coexisting along the HH 223-A to -F emission region.
At both positions the emission appears blueshifted, although no
A trend for the velocity can be found, consisting in an increasesignificant variation in velocity along the knot is found. In addition,of the absolute value of the velocity, suggestive of an acceleration,
we detected emission from other H2 lines, both in the H and K bands,
the behaviour being consistent with that of the 2.122 µm emission
from east to west, towards the location of SMA 2. The general trend(see e.g. the PV map of the 1.748 µm emission in Fig. 6). Note,
seems to be broken at a few positions (e.g. at A1 and D knots).
in contrast, the spatial displacement found between the [Fe II] andHowever, this might be expected because of the highly complex
H2 emissions, inferred from their PV maps (Fig. 4). The position ofkinematics of HH 223-A-F already found at optical wavelengths,
Figure 12. Image of the L723 field through the H2 2.122 µm line filter (continuum is not subtracted). The arrows indicate the proper marcsec) with thethe [Fe II] emission peak is coincident (within 0.2
where a fully spatial sampling of the emission was obtained with
derived for each knot. The position of SMA 2 is indicated by a × sign. The scale of the arrows (in km s−1 ) is indicated by the double
it isoffsetpeak of HH 223-B-east, the H2 weaker substructure,
IFS (López et al. 2012). One indication of such a highly complex
bottombut
rightcornerof the map.
by 1.4 arcsec from HH 223-B-peak.
kinematics can be found by examining the PV maps of the 2.122
The slitlet S4 included emission from the intensity peak of theµm line of HH 223-A-F at spatial scales of ∼1 arcsec: as can beto the west of
inhomogeneous environment, have been observe
im-millimetre sources, are also misaligned with rebow-shaped feature labelled HH 223-C in the narrow-band
H2theseen from Fig. 6, a change in velocity can be appreciated along
spectofto itthe(labelled
east–west CO outflow axis. The CO velocity maps of
HH 110, see Kajdic et al. 2012).
age, and from a weaker condensation, south-east
HH 223-A (i.e. in S2). Moving from east to west through the slitlet,
Leework).
et al. (2002)
show the presence of a clump of blueshifted CO
HH 223-C-North and C-East, respectively, in thisA spatial
we found an intensity peak of emission, labelled as knot HH 223-A1
emission,superposedontheredshiftedCO
outlfow
lobe,
aroundshift between the [Fe II] and H2 emissions was already found in thein Table 4, and another secondary emission enhancement (HH 2233.4 Full spatial velocities
location ofinthese
H and K narrow-band images, and also can be theappreciatedthe H2 knots. It might be indicative of a complex
A2 in Table 4), their peaks being separated ∼1 arcsec. From the
The full spatial velocity (Vtot ) and the angle (φ)
emission isbetween
offset molecular clumps at these positions that dePV maps of this work. The peak of the [Fe II]interaction
spectra obtained for these two HH 223-A knots, we found a differviates theproper motions away from the outflow axis. Such
motion and the sky (with φ ≥ 0 towards the o(HHknot
223-Cby ∼1.2 arcsec west from the peak of the H2 emission
ence in velocity of ∼50 km s−1 between them, being the eastern
deviations, attributed to complex interactions of the outflow with antained for the knots for which the two velocity
knot more blueshifted. It is worth noting that HH 223-A presents
North).
two emission peaks separated ∼1 arcsec, partially resolved in the
The slitlet S5 included emission coming from the bow-shaped
Hα images (López et al. 2009). The radial velocities derived fromfeature HH 223-E, and from HH 223-D. We foundan 447,
offsetof
MNRAS
2588–2602(2015)Knot
HH 223-

CO outflow軸の方向にknotsは固有運動を持っている。固有運動は双曲流パタンに従い、knotsは東西で逆方向の運動を持つ。!

!

Summary;!
HH223 H2 outflowの3D力学構造を導出した。その駆動源はYSOのSMA 2であり、最も近いH2 knotsはHH223-K1である。力学構造はH2とCo outflowは関係していることが示され

MNRAS 447, 2588–2602 (2015)た。
ページ 7/10

ck, D. K. Ojha, M. Tamura, H. Linz, M. R. Samalhosh1,5

nomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India
al Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
rsité, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, F-13388 Marseille, France
29.
morphology
and411stellar
content of the Galactic H II region IRAS 16148−5011!
Radio Astrophysics,
Tata“Study
Institute of of
Fundamental
Research, Pune
007, India”!
K. K. Mallick, D. K. Ojha, M. Tamura, H. Linz, M. R. Samal and S. K. Ghosh!
mber 4. Received 2014 December 4; in original form 2014 September 18
http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stu2584!

TION

2308

ABSTRACTK. K. Mallick et al.

An investigation of the IRAS 16148−5011 region – a cluster at a distance of 3.6 kpc – ispresented here, carried out using multiwavelength data in near-infrared (NIR) from the 1.4 m
Infrared Survey Facility telescope, mid-infrared (MIR) from the archival Spitzer GLIMPSE(Galactic Legacy Infrared Midplane Survey Extraordinaire) survey, far-infrared (FIR) fromthe Herschel archive, and low-frequency radio continuum observations at 1280 and 843 MHzfrom the Giant Metrewave Radio Telescope and Molonglo Survey archive, respectively. A
combination of NIR and MIR data is used to identify 7 Class I and 133 Class II sources in theregion. Spectral energy distribution (SED) analysis of selected sources reveals a 9.6 M⊙ highmass source embedded in nebulosity. However, Lyman continuum luminosity calculation
using radio emission – which shows a compact H II region – indicates the spectral type of
the ionizing source to be earlier than B0-O9.5. Free–free emission SED modelling yields– associated with the central nebula ) for our work. The compilathe electron density as 138 cm−3 , and thus the mass of the ionized hydrogen as ∼16.4 M⊙ .
Ks,tionH,
JのColourimageof Lumsden
et al. (2013)composite
also reveals other nearby
regions at
this distance, providing preliminary indications that this could be
Thermal dust emission modelling, using the FIR data from Herschel and performing modified
a part of larger complex. In the early analyses of Haynes, Caswell& Simons (1979, also see Chan, Henning & Schreyer 1996), ra2310
K. K. Mallick et al.
blackbody fits, helped us construct the temperature and column density maps of the region,
dio continuum emission at 6 cm was detected, with peak in theneighbourhood (!1 arcmin, positional accuracy ∼30 arcsec) of
22−2
this region, and it was predicted to be harbouring massive young
which show peak values of 30 K and 3.3 × 10 cm , respectively. The column density maps
stellar objects (YSOs). The IRAS colour analysis by MacLeod et al.
(1998) was also found to be consistent with that for an H rereveal an AV > 20 mag extinction associated with the nebular emission, and weak filamentarygion, and Molinari et al. (2008) – using the IRAS ‘[25−12]’ colour
value – have suggested that this region is among the younger IRASdetected regions. A high-mass stellar source was detected by Grave
structures connecting dense clumps. The clump associated with this IRAS object is found to& Kumar (2009) with the help of spectral energy distribution (SED)
fitting using near-infrared (NIR) to millimetre (mm) data. Analyses
have dimensions of ∼ 1.1 pc × 0.8 pc, and a mass of 1023 M⊙ .
of this region at 1.2-mm dust continuum emission and at molecFigure 1. Colour composite image of the IRAS 16148−5011 region, madeusing the Ks (red), H (green), and J (blue) band images. The area of analysis
in this paper has been marked by the dashed rectangle. Plus symbol marks
the IRAS catalogue position of IRAS 16148−5011. The scale bar shows 1 pc
extent at a distance of 3.6 kpc.
1

IIular lines have revealed the presence of dense gas, with a large
column density, as well as massive clumps. The total luminosity
estimate (∼4.4 × 104 L⊙ , by integrating IRAS flux densities) has
also been found to be well in the regime of high-mass stellar objects(Fontani et al. 2005; Beltrán et al. 2006). Therefore, taking into
account these characteristics, this region makes a good candidate
to carry out an investigation for understanding the morphology andstellar population. However, since this is a possible H II region with
an embedded cluster, multiwavelength observations are required to
fully discern this region’s various constituents and how they relate
to each other. In this paper, we have tried to accomplish this using
deep NIR observations, archival mid-infrared (MIR) Spitzer data,archival far-infrared (FIR) Herschel data, and low-frequency radio
continuum observations.

Key words: stars: formation – dust, extinction – H II regions – ISM: individual objects:
IRAS 16148−5011 – infrared: ISM – radio continuum: ISM.

!

対象；IRAS 16148-5011, more
3.6kpc,!
so as high-mass star formation is not very well understood(Zinnecker & Yorke 2007).
観測；NIR (Infrared Survey
Facility telescope 1.4m)!
n activity is known to take place in clusters
IRAS 16148−5011 is an infrared nebula (Fig. 1) in the south3), and as such, !observational
studiesof young
ern sky (α2000 = 16h 18m 35.s 2, δ 2000 = −50◦ 18′ 53′′ ) associated with
MIR
(Spitzer
GLIMPSE)!
which is an IR cluster found using the Two Micron All-Sky Survey
uster regions are imperative, because they serve!
FIR (Herschel archive)!
her investigate various associated processes and(2MASS) data by Dutra et al. (2003). It is located at the Galactic!
low-frequency843
ue to the youth of such regions
and the fact that 1280,
plane(l ∼MHz!
333.◦ 047, b ∼ +0.◦ 037) and is in the vicinity of the well2314106.
K. K.
Mallick etotheral. star-forming
known star-forming region RCW
Thoughhas still not been dispersed, these clusters can beregions are present nearby, IRAS 16148−5011 appeared to be a
arious theories related to star formation, stellar
relatively isolated region in past mappings (Karnik et al. 2001;
as well as stellar and cloud evolution. Of even
<Conclusion>!
Mookerjea et al. 2004). Kinematic distance estimates to this rere the cluster regions
which harbour high-massClass
I (7),
Class
gion 同定し、SEDから星雲に埋もれた9.6vary from ∼3.3 to 11.9 kpc (near- and far-distance estimates;
se such regions 1.
are few
and farbetween,and II (133)
Molinari et al. 2008), and we adopt the distance of 3.6 kpc from
Msunの大質量星を見つけた。!
Lumsden et al. (2013, based on the spectrophotometric distance of aprominent source – ‘G333.0494+00.0324B’ in their nomenclature.res.in

1 see http://rms.leeds.ac.uk/.

In S
spondithe steof the r
are pre2 OB

2.1 NI

NIR p
and Kδ 2000 ∼1.4 m I
observ
Imagersimultatelescoeach oof view
obtaineFive seditherewere oin eachsize of(∼10 aP9172
Foll
bad pix
traction
point sALLST
13 southe ins
P9172.
and thecatalog
tude ≤
Ks mag
The fin
The arecompa
δ2000 ∼
Com
ing out
A fixedsis carr
numbe
complecomplethe J, H

2.2 RaRadio
2012 N(GMRT
in an aing a d26.2 artains 1MNRAS 447, 2307–2321 (2015)

Figure 2. (a) Histogram of the IRAC spectral indices. The limits for Class I and Class II sources have been marked by dashed grey vertical lines. (b) NI
CCD using the procedure of Gutermuth et al. (2009). The regions of Class I and Class II sources have been labelled. The identified YSOs have beenwith black solid circles. (c) NIR CCD in CIT system. The red curve and the blue line show the dwarf locus (Bessell & Brett 1988) and the CTTS locus
et al. 1997), respectively. The grey dot–dashed line is the extension of CTTS locus. Three parallel slanted dashed lines mark the reddening vectors, draw
extinction laws from Cohen et al. (1981). Three separate regions, ‘F’, ‘T’, ‘P’ have been labelled on the plot. The source marked with a cross in ‘P’ reghigh-mass YSO (see text).

!
!

(CTTS) locus from Meyer, Calvet & Hillenbrand (1997). All the locicurves as well as sources’ colours were converted to CIT (California
Institute of Technology) photometric system (using Carpenter 2001)
for this analysis. The slanted dashed lines are the reddening vectors,
drawn using the reddening laws of Cohen et al. (1981) for the CIT
photometric system. In this CCD, three separate regions have beenmarked, similar to Ojha et al. (2004a,b). The sources in ‘T’ and‘P’ regions are taken to be Class II sources (Lada & Adams 1992),
as they exhibit IR-excess emission. In the ‘P’ region, since there
could be a slight overlap between Herbig Ae/Be stars and Class II
sources (Hillenbrand et al. 1992), we conservatively took only thosesources which were above the CTTS locus extended into this region(similar to Mallick et al. 2014). It should be noted that sources inthe ‘T’ region could also contain a few Class III sources with small
IR-excess.

!

Finally, we merged the sources identified in each step. An
lapping source might have different identifications in differenThus, in the final list, the class of a YSO was taken as that in w
was identified as first in the above order of steps. A total of 7
and 133 Class II sources were obtained (for a FoV of ∼cmin × 5.5 arcmin encompassing the molecular cloud, as m
in Fig. 1) in the final YSO catalogue, which is given in Tabl
3.2 SED of YSOs

SED modelling was carried out for (a subset of) YSOs to gmates of their physical parameters. The grid of YSO model
Robitaille et al. (2006) – implemented in the online SED fittiof Robitaille et al. (2007) – was used for this purpose. The
model consists of a pre-main sequence (PMS) star surroun

MNRAS 447, 2307–2321 (2015)

2. 低周波電波放射はcompact HII regionを明らかにし、その放射は南University Press on behalf of the Royal Astronomical Society

側に伸びている。

Lyman continuum luminosityの計算は、ionizing sourceとして(単一Figure 7. Spitzer 8.0 µm image of the region with overlaid surface density
contours (at 5, 5.75, 6, 7, 7.5, and 8 YSOs pc−2 ) in cyan, and visual extinction
contours (at AV = 4, 4.5, 5, 6, and 6.5 mag) in blue. Green plus symbol
marks the IRAS catalogue position of IRAS 16148−5011, green cross thehigh-mass source (#16 from Table 2), while the diamond and box symbols
mark the mm and MSX peaks, respectively, from Molinari et al. (2008).

のソースを考えたとき)Spectral typeのlower limitとしてB0-O9.5 となった。
HII regionの力学年齢は0.3–0.5 Myr

and MSX peaks from Molinari et al. (2008, 16148-5011 MM 1 and16148-5011 A from their tables 2 and 3, respectively) have also beenmarked on the image. The marked mm peak is almost coincident
with the (mm) peak from Beltrán et al. (2006, 16148-5011 Clump 2
from their table 2). The surface density and extinction contours were
calculated as follows. The surface density analysis was carried out
using the nearest-neighbour (NN) method (Casertano & Hut 1985)
to discern the YSO clusterings in the region. We chose 20 NN,
similar to Schmeja, Kumar & Ferreira (2008), Schmeja (2011), andMallick et al. (2013). The extinction in the region was estimated byconstructing the extinction map with the help of the NIR photometric data. We use the NIR CCD (see Fig. 2c) for this purpose. In thisNIR CCD, the ‘F’ region mostly contains the main-sequence fieldsources along with a few probable Class III sources. Since theseare sources which have almost lost their circumstellar material, any
extinction they exhibit will come from interstellar – rather than circumstellar – material. The sources from this ‘F’ region, which were
not identified as a YSO in Section 3.1, were therefore selected. Subsequently, we used a method similar to the Near-IR-Colour-Excess
method of Lada et al. (1994), but here the (H − K) colour excess
was estimated by dereddening the sources – along the reddening
vector – to the low-mass end (turnover onwards) of the dwarf locus(which was approximated as a straight line). AV was then calculated

SEDのmodelingによるFree-free emissionは電子密度138/ccが得ら!

れ、電離されたHIIガス〜１６．４Msun 質量である。!

ページ 8/10

Figure 8. Spitzer 8.0 µm image of the region with overlaid 1280-MHz contours (at 3σ , 4σ , 5σ , 7σ , 10σ , 12σ , 15σ , 20σ , and 22 σ , where σ ∼ 8.85 mJy;
resolution ∼56 arcsec × 35 arcsec) in cyan and 843 MHz contours (at 5σ ,7σ , 10σ , 15σ , 20σ , 30σ , 40σ , and 50 σ , where σ ∼ 5.57 mJy; resolution ∼ 56 arcsec × 43 arcsec) in blue. The symbols are same as for Fig. 7.

2316

K. K. Mallick et al.

the image, both the surface density as well as extinction contours
are in the southern portion of the nebular emission and appear to be that thalong the sharp boundary of the nebula. The highest surface density H II recontour levels coincide with the highest extinction levels. Though
we would have expected to see high interstellar extinction along the 4.3 C
main body of the nebula, this is not so possibly because the NIR
Fig. 1
observations are not deep enough to detect stars from behind the
IRAS
nebula and thus the extinction of the nebula should be higher than and 8.
the highest extinction contour level here (it is later estimated to be 8.0 µm
AV > 20 mag; see Section 4.4) It should also be noted that the cluster the w
detected here is in the southern part of the nebula also coincident ture a
4.5 µmwith extended radio continuum emission (see Section 4.2).

shock
and C
uum e
et al. 2
Fig. 8 shows a Spitzer 8.0 µm image of the region with over- The
laid 843 MHz contours from MGPS and 1280 MHz contours from high-m
GMRT. Rest of the objects marked are same as in Fig. 7. The central MSX
Figure 10. The fitted free–free emission model for the H II region. The datacore
appears to be a compact H II region, near whose peak lie the (green
points at 843 and 1280 MHz have been marked with solid circles.
mm peak and the high-mass source. The 843 MHz contours show type (tableextended
the centralcompact
region.
of 2.6 ×emission
10−13 cm3ins−1addition(Stahler to& Palla2005). For
no , we use
the The ≥ B0−3 in the southern part and none in the northextended
cmonlyfrom Beltrán et al. (2006). R(t) is taken as
value ofemission
4.8 × 104 is(assumgeometric
mean of H
fittedGaussiansource sizes
Tableof3.dense have e
II region,
indicating
the frompresenceernthepartof the compact
Using theseformulaeandnorthern
the 1280 MHz
weis
calculatedS∗ andto the as O8
molecular
cloudin thepart data,which
not ionized47.41−1
photons s and ∼0.3 Myr, respectively. If we weret as ∼10

4.2 Radio morphology

3. 星雲の中心領域で(PDRをトレースする)PAHのring状の構造がcompact HII regionの境界近くで見られた。
３つのNIR,MIR,visible 中心天体がある。6.2, 9.60, 8.5 Msun.

!
!
!
!
!
!
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!
!
!
!
!
!
!
!
!
!
!
!
!

電波のfluxとMIR SED fittingの不一致から、この領域に埋もれた大質量星が存在し得る!
IRAS 16148−5011

2317

Figure 11. 1 arcmin × 1 arcmin central region of IRAS 16148−5011. The images (left-to-right, top-to-bottom) are J, Ks , 3.6, 4.5, 8.0, and 24 µm. North isup and east is to the left. The symbols on the Ks band image are same as Fig. 7 (green cross is #16 from Table 2). The numbered sources on the 3.6 µm image
are from Table 2. The cyan arrows on the 8.0 µm image mark the ‘ring-like’ morphology. The white patch on the 24 µm image is the saturated region.

effects. Volk et al. (1991), on the basis of low-resolution spectra, hadclassified IRAS 16148−5011 as a PAH source. Bulk of the radio
emission (see Fig. 8), as well as the 24 µm emission from hot dust,
is confined within this ring-like feature as expected (Watson et al.
2008). Some of the emission from this ring-like structure could also
be due to swept-up material by the expanding ionization front of
the H II region.

rejecting the pixel values outside ± 2σ in each iteration, till thefit converged. The same patch of the sky was used for each band.
The background flux level, Ibg , was thus determined as 0.29, 2.67,
1.27, and 0.45 Jy pixel−1 for the 160, 250, 350, and 500 µm images,
respectively. We note that the 70 µm image, though available, was
not used here as the optically thin assumption might not hold trueat this wavelength (Lombardi et al. 2014).
Modified blackbody fitting was subsequently carried out on a
pixel-by-pixel basis using the following formulation (Battersby
et al. 2011; Sadavoy et al. 2012; Nielbock et al. 2012; Launhardt
et al. 2013):

4. Thermal dust emission modeling (Herschel dataより、黒体放射fit)で、温度と柱密度が得られる。peakは30K, 3.3 x 10^22 /cm^2。

柱密度マップは、星雲放射に付随したAv>20 magの減光を明らかにし、またIRAS
4.4 Herschel results

16148-5011とdense
clumpsに接続したweakfilamentary structureがあった。
Thermal emission from cold
dust lies in the FIR wavelength range,
and thus its analysis can be used to obtain the physical parameters

!
!
!
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!
!
!
!
!
!
!
!

Sν (ν) − Ibg (ν) = Bν (ν, Td )#(1 − e−τ (ν) ),(8)τ (ν) = µH2 mH κν N (H2 ),(9)like dust temperature and column density of a region (Battersby
IRAS天体に付随したクランプは~1.1pc
x 0.8pc,
1023Msunである!
with
et al. 2011; Launhardt et al. 2013). This was carried out by the
K. K. dust
Mallick
et al. whose Rayleigh–SED modelling of 2318
the thermal
emission,
Jeans regime is covered by the Herschel FIR bands (160–500 µm),
using the following order of steps.
First of all, the surface brightness unit for all the images was converted to Jy pixel−1 . Since the PACS image is already in Jy pixel−1 ,this step was only required for the SPIRE images (whose units are in
MJy sr−1 ), and was carried out using the pixel scales for the respective SPIRE bands. Next, the 160–350 µm images were convolved
to the resolution of the 500 µm image (∼36 arcsec, lowest among
all images) using the convolution kernels of Aniano et al. (2011),
and regridded to a pixel scale of 14 arcsec (same as 500 µm image).
Using these final reworked images with same resolution and pixel
scale, a background flux level, Ibg , was determined from a ‘smooth’(i.e. no abrupt clumpy regions) and relatively ‘dark’ patch of thesky. The distribution of individual pixel values in the dark patch of
the sky, for each of the bands, was fitted with a Gaussian iteratively,

where, ν is the frequency, Sν (ν) is the observed flux density, Ibg (ν)
is the background flux in that particular band (estimated above),Bν (ν, Td ) is the Planck function, Td is the dust temperature, # is thesolid angle (in steradians) from where the flux is obtained (just thesolid angle subtended by a 14 arcsec × 14 arcsec pixel here), τ (ν) is
the optical depth, µH2 is the mean molecular weight (adopted as 2.8
here), mH is the mass of hydrogen, κ ν is the dust opacity, and N(H2 )is the column density. For opacity, we adopt a functional form ofκ ν = 0.1 (ν/1000 GHz)β cm2 g−1 , with β = 2 (see Hildebrand 1983,
Beckwith et al. 1990, André et al. 2010). For each pixel, equation (8)was fitted using the four data points, keeping Td and N(H2 ) as freeparameters. Pixels for which the fit did not converge, or the error
was larger than 10 per cent, had their values taken as the median ofeight immediate-neighbour pixels. The final obtained temperature
MNRAS 447, 2307–2321 (2015)

ページ 9/10
Figure 12. (a) Dust temperature map, and (b) column density map of the region around IRAS 16148−5011, derived using the SED fitting to the thermal dust
emission.

!

The Astrophysical Journal Letters, 798:L44 (4pp), 2015 January 10

doi:10.1088/2041-8205/798/2/L44⃝
C 2015. The American Astronomical Society. All rights reserved.

30. “SHADOWS CAST BY AWARP IN THE HD 142527 PROTOPLANETARY DISK” !
S. Marino, S. Perez, and S. Casassus!
SHADOWS CAST BY A WARP IN THE
HD 142527 PROTOPLANETARY DISK
http://stacks.iop.org/2041-8205/798/i=2/a=L44?
key=crossref.ce45c878a03a5d94cabe21b0186f7bac!
S. Marino1,2 , S. Perez1,2 , and S. Casassus1,2!
2015,1ApJL
798,deL44!
Departamento
Astronomı́a, Universidad de Chile, Casilla 36-D Santiago, Chile; smarino@das.uchile.cl
2 Millenium Nucleus “Protoplanetary Disks in ALMA Early Science,” Universidad de Chile, Casilla 36-D Santiago, Chile（チリ大の人たち）!
Received 2014 December 14; accepted 2014 December 15; published 2015 January 7
ABSTRACT
Detailed observations of gaps in protoplanetary disks have revealed structures that drive current research on
circumstellar disks. One such feature is the two intensity nulls seen along the outer disk of the HD 142527 system,
which are particularly well traced in polarized differential imaging. Here we propose that these are shadows castby the inner disk. The inner and outer disk are thick, in terms of the unit-opacity surface in the H band, so that theshape and orientation of the shadows inform on the three-dimensional structure of the system. Radiative transfer
predictions on a parametric disk model allow us to conclude that the relative inclination between the inner andouter disks is 70◦ ± 5◦ . This finding taps the potential of high-contrast imaging of circumstellar disks, and bears
consequences on the gas dynamics of gapped disks, as well as on the physical conditions in the shadowed regions.
Key words: planet–disk interactions – protoplanetary disks – stars: individual (HD 142527)

天体HD142527の原始惑星系円盤のinner (とouter disk)の位置関係等のmodelingをした。!

Motivated by our research on the intra-cavity gas kinematics(Casassus et al. 2013a; S. Casassus et al. 2015a, in preparation),
we considered the possibility of a continuous warp linking
different orientations between the outer and inner disks. The
by observational breakthroughs, such as the direct images ofunderlying parametric disk model is documented in Section 2,
protoplanetarygaps in near-infrared scattered
light. A particu・H-bandのPDI(Polarized
differential
imaging)で二箇所のnullが北(11.5時方向)と南(7時along with the radiative transfer setup we implemented tolarly interesting target is the Herbig Ae/Be star HD 142527, at
calculate the emergent specific intensity fields. Our results are方向)に存在する。Figure
2 (a)!
a distanceof ∼140 pc. The disk surrounding
HD 142527 boasts
presented in Section 3. While the inner disk intercepts stellar
the largest inner cavity known, with a 1′′ radius and seen at a・連続波の観測ではそのPDIで抜けている場所でdust
densitylight, shadowsare cast peakがあり、物がない訳
on the outer disk such that their shape andclose to face-on
orientation, with an inclination of about 24◦
orientationaresensitive
measures of the inner disk orientation.
(Fukagawa et al. 2006; Fujiwara et al. 2006). Improvements in
ではない(sub-milli波観測？)!
We briefly consider the consequences of this finding in Section 4.
Ks -bandcoronography revealed fine structures in the outer disk,
including＜内側円盤＞!
two intriguing intensity nulls (Casassus et al. 2012)
2. PARAMETRIC MODELING
that break the continuity of the outer disk. In this Letter, we
propose that
these
are
shadows
cast
byawarpedinner
disk.
・VLTIの観測で内側円盤があることが分かっている。が配置関係などは分かっていない!
2.1. Underlying physical structure
Recent polarized differential imaging (PDI) of HD 142527
We constructed a synthetic model of HD 142527 inspiredat the H and Ks bands revealed a striking view of its dustby multiwavelength observations and gas kinematics, and dedepleted
gap, outer disk, and spiral structures (Canovas et 1のような状況を主張している。内側円盤が外側
al.
結果として、彼らの主張としてはFigure
tailed enough to provide a comparison point with the near-IR
2013; Avenhaus et al. 2014; Rodigas et al. 2014). These H-band
scattered light images as well as the radio continuum emispolarized
intensity images outline the outer-disk nulls with the degでP.A.=-8±5degをもち、内側円盤が外側円盤に対して相対的なinclinationとして70±5
sion and gas kinematics. As explained in a companion paper
best available detail thus far. One null is found ∼11.5 hr (north(S. Casassus 2015b, in preparation), this synthetic modelof the 円盤に影を落としている。
star), and the other is found at 7 hr (south). The northern
matches the observed
spectralenergy distribution. Here we docnullTheisAstrophysicalparticularly
puzzling
it coincides
with10the location of
JournalLetters,as
798:L44(4pp), 2015 January
Marino,Perez, & Casassus
ument
the
aspectsofthereportTheintoAstrophysical Journal
Letters,
798:L44(4pp), 2015
January
10model that are relevant to this Marino,
Perez, & Casassus
the dust density peak, whose spatial distribution is shapedwith Rs = 148 AU and where the azimuthal contrast in surface
H-bandimages.
a large crescent (Ohashi 2008; Casassus et al. 2013b; Fukagawa
density is set to onc =the10.0.
The volume
density follows with astandard
distribution:
The model
consists of a star at the center of a dusty disk
et al. 2013). Thus, the northern null cannot be interpreted
as a vertical Gaussian)*
separatedlack of material.
φ) three z2zones with different density structures,Σg (r,intoz, φ) grain= √ sizeexp−,
distributions,and (5)compositions: an inner disk,
As witnessed by a bright thermal IR central point source, ρg (r,dust
2
2H
2π H
a gap, and an1.17outer disk. A schematic render of the model iswell in excess of the photospheric emission, the large dustwith H (r) = 20.0(r/(130. The1.exact
value ofthe
thisstar using a Kurucz template
providedAU))
in FigureWe modeldepleted cavity of the HD 142527 disk is in fact a gap (Fujiwara
flaring exponent is not well constrained.
spectrum(Castelli&
Kurucz2003)
et al. 2006; Verhoeff et al. 2011; Rameau et al. 2012). While
This parametric model also includes the effect of dust at a temperature of 7750 K
andthewith
a radiusof 3.3
with the extinction
knowledge of the outer edge of the gap is limited by theorytrapping,rather following
procedure
described
in R
Birnstiel
et consistental.
⊙ to beand Pinillaofet A
al. (2012).
However, the bulk of the opacity
than observational detail, the inner regions of the gap are (2013)
poorlyR = 1.3 ± 0.3 reported in Close et al. (2014).
in the H band is driven by particles well below the sizes required
The inner
disk
0.3 AUand cuts off at 10.0 AU. It is
resolved. A ∼10−7 M⊙ yr−1 stellar accretion rate (Garciafor
Lopez
efficient aerodynamic
coupling,
andstarts
so the at
effectsof dustdescribed
surfacedensity
to r −1 and a scale
et al. 2006) would quickly deplete the inner disk, whose
dust in the outer
trapping
disk areby
notarelevantto thisreport.proportionalThe−9detailed in Section
disk isopticallyheight3 confirmof 1.0 that
AUthe
atouter
10 AU
with
a flaring exponent of 1.1. It is
mass is estimated at ∼10 M⊙ (Verhoeff et al. 2011). Thisruns
innerthick at the H band and that the scattered light does not trace thecomposedofamorphous
carbonand silicate grains with sizes
disk can be thought of as a transient feature of stellar accretion:
crescent shape seen in the submillimeter.
Schematic view with arbitrary orientation of the parametric model
between 0.1 to 2.5 µm, with a total dust mass of 5.0 × 10−9 M⊙ .
it isFigure
the1.convergencepoint of matter being accreted from thepresented in Section 2. The central star is placed at the origin. The outer disk lies
Emergent an
intensities
in thex–y plane. Thein
angle
the relative
inclination
between theof
We2.1.1.
introduce
inclination of α = 70◦ to this region with respectmass
reservoir
theα isouterdisk(Casassus
etmidplaneal. 2013a,
2013b;
Figure 2. Impact of the inner3 disk orientation on the H-band light scattered off the outer disk. (a) NACO-PDI H-band image from Avenhaus et al. (2014) compared
the outer disk and the plane of the inner disk. The dust mass density distribution
Wetheuseradmc3dradiative
transfercomputations(verwith
C O(2–1)
emission
atfor
systemic
velocity
fromPerez
et al.
(2014). The C O(2–1)
emission, represented here as one white contour at 0.75 maximum, shows
to theouter
disk
midplane.
S. Casassus
etandal.outer
2015a,
in preparation).
Long-baseline
opticalof the inner disk
disk sections
are rendered in false color.
The gap isthat thepositionangle (P.A.) of the
outer
disk
is at −20
east of north,andand perpendicular
to the solid gray double arrow, while the position angle of the intensity nulls
sion
0.38;
etarrowal.
2014).
Scatteringpolarization
shown devoid of material
for simplicity.
The inner disk
is scaled
up indisk,
size andand foundThe
gap
120transfer
AUprediction
fromfor 10.0
isH band
described
bydiskaP.A.s (indicated in
is indicatedthe dashed double(−8 ).spans(b)–(f) Radiative
polarizedAU.
intensity It
in thefor different innerinterferometry(VLTI)
has resolved
this
inner
it byDullemondforthe onlast
are treatedwith scatteringmatricesforouterourdisks. The x− and y−axes indicate offset along R.A. and decl., in arcsec.
density for better visualization.
degreesthe scatteringplots) and for different
relative inclinations
α between the
inner and thesurfaceto r −1inand a scale height of 18 AU
to be highly chrystalline (van Boekel et al. 2004). However,its dust species,different
eachdensity
one with aproportional
power-law distribution
at exponent
100theAU
a parallel,
flaring
exponent
1.6. 1969).
It isChristiaens
composedof propose thatgrainwith anof with
−3.5. InTo
computethefull(Hunter
dust oforientationand extent remain ellusive.
& Toomre
et al. (2014)
so thatsizes
it is inconsistentwithobservations.
for low
amorphous carbon and silicate grains with sizes between 1.0 to
1. INTRODUCTION
観測事実として、!
The studyof planet formation is rapidly being revolutionized＜外側円盤＞!

!

18

18

◦

10.0 µm. The total dust mass of this section is 1.0 × 10−8 M⊙ .
The ページ
disk in thissection10/10 connects the inner and outer regions
varying the inclination linearly from 70◦ to 0◦ between 10 and
15 AU, where it matches the outer disk orientation. A larger
warp would have been obvious in the 12 CO kinematics inside
the gap (Perez et al. 2014), with a concomitantly larger region
where the inclination crosses the plane of the sky.

◦

the two-armed spirals in the outer disk of HD 142527 might beand negative
α, the inclinationand orientation
theofshadows
opacityand scatteringmatrices,
we madeofuse
complementary
indicative of a recent close stellar encounter (see also Quillen
do notinfitRADMC3D
the shape of the(seefrom
Figures2(c) and(f)). Acodes
andnulls
a codeBohren& Huffman(1983)
1
et al. 2005). Although a flyby could also explain the tilt between
qualitative match with the observations is obtained with an innerfor“Miesolutions”◦ scattering by homogeneous spheres. We
the inner and outer disks, no partner for such a stellar encounterdisk
inclined
at α = 70torelative to the outer disk, and along a◦used
the−8opticalamorphous carbon
grainshas been identified.
P.A. of(Figureconstant2(b)). We tables
can ruleforout configurations
with
◦
◦
◦
In the prototypical T-Tauri disk TW Hya, Rosenfeld et al.
P.A.s Libeyond10 of −8 sothat theand1σ errorbar is about
5 . we usedfrom
& Greenberg(1997),
for silicate
grains
◦◦(2012) proposed a warp to understand the sinusoidal (m = 1)
Likewise,therelative
inclinationisconstrained
within60–80Henning & Mutschke (1997).
◦
azimuthal
soWethatimplemented
the 1σ error is also
∼5.
our model in RADMC3D
using spherical modulation seen in Hubble Space Telescope images,◦
An inner disk orientation along a P.A. of ∼60 has been

as well as features of the CO gas kinematics of the innermost