The inner rim of 51 HAeBe
protoplanetary disks.
A VLTI-Pionier survey
Lazareff et al.
ESO Large Program #090C-0963
Astronomy & Astrophysics manuscript no. HAeBeLP_master
May 11, 2016
c ESO 2016
Few puffed-up rims found in large interferometric survey of Herbig
AeBe stars
A survey of Herbig AeBe stars using PIONIER-VLTI
B. Lazare↵1 , J.-P. Berger2, 1 , J. Kluska3 , J.-B. Le Bouquin1 , M. Benisty1 , F. Malbet1 , C. Koen4 , C. Pinte6 , W.-F. Thi7 ,
O. Absil5 , F. Baron8 , A. Delboulbé1 , G. Duvert1 , A. Isella10 , L. Jocou1 , A. Juhasz9 , S. Kraus3 , R. Lachaume11, 12 ,
F. Ménard1 , R. Millan-Gabet14, 15 , J. Monnier13 , K. Perraut1 , F. Soulez16, 17 , M. Tallon16 , E. Thiébaut16 , W. Traub18 ,
and G. Zins19
1
2
3
4
5
Univ. Grenoble Alpes, IPAG, F-38000 Grenoble, France CNRS, IPAG, F-38000 Grenoble, France
ESO Karl-Schwarzschild-Strasse 2 D-85748 Garching bei München, Germany
University of Exeter Department of Physics and Astronomy Stocker Road, Exeter, Devon EX4 4QL UK
Department of Statistics, University of the Western Cape, Private Bag X17, 7535 Bellville, South Africa
Département d’Astrophysique, Géophysique et Océanographie, Université de Liège, 17 Allée du Six Août, 4000, Liège, Belgium
The structure of a disk inner rim
AA48CH07-Dullemond
ARI
16 July 2010
20:8
Dullemond & Monnier 2010
Questions:
❖
❖
Accretion
shock
Rounded off dust inner rim:
a dust chemical reactor
Dust-free inner
gas disk
Radial structure
Vertical structure
❖
Dust temperature
❖
Dust composition
❖
Non axi-symmetry
Optically
thick gas (?)
Annu. Rev. Astro. Astrophys. 2010.48:205-239. Downloaded from www.annualreviews.org
by European Southern Observatory on 07/16/14. For personal use only.
❖
Magnetospheric
accretion
Shadow cast
by the gas (?);
a possible “safe
haven” for dust
Near-IR-emitting
surface of the
dust inner rim
Weak shadow cast
by the dust rim (?)
Figure 3
Pictographic representation of the inner disk region out to a few astronomical units. Shown are the
magnetospheric accretion depicted near the star, the dust-free gas disk in the middle, and the dust rim on the
right.
Goldreich model and thus obtained a complete description of the SEDs of Herbig Ae/Be stars in
terms of a simple irradiated disk model.
So if this simple model of the NIR bump is basically correct, then one may wonder why mainly
Herbig Ae/Be stars show such a huge bump while T Tauri stars are not known for displaying
such a conspicuous feature. Dullemond, Dominik & Natta (2001) argue that because the stellar
luminosity is at much longer wavelengths for T Tauri stars, a bump of this kind would be partly
“swamped” by the flux from the star, though a close look at the spectrum should still reveal such a
bump. On first sight, T Tauri star SEDs do not show such a strong bump. But through a careful
subtraction of the stellar spectrum, Muzerolle et al. (2003) show that T Tauri stars consistently
have such a NIR bump, though perhaps weaker in a relative sense than the Herbig stars. So in
that sense, T Tauri stars are no different from Herbig stars.
In spite of the early success of these models, there was no easy way of telling with just NIR
photometric data whether it was indeed the true nature of these objects. Indeed, much simpler spherically symmetric envelope models, in which the dust was also removed inward of the
dust evaporation radius, could also fit the NIR bump and even in a number of cases the entire SED (Pezzuto, Strafella & Lorenzetti 1997; Malfait, Bogaert & Waelkens 1998; Miroshnichenko et al. 1999; Bouwman et al. 2000; Vinković et al. 2006). In fact, such models appear
to be more consistent with the lack of clear observed correlation between the NIR flux and
the disk inclination. For a simple perfectly vertical wall model of the rim such a correlation
is clearly expected, with little NIR flux observed at near face-on inclinations as illustrated in
Figure 4a,b, and discussed in more detail in Section 3.1. Perhaps the most clear counter-example
is AB Aurigae, which has a huge NIR bump (see Figure 2) but is known not to be very far from
face-on (e.g., Eisner et al. 2003; Corder, Eisner & Sargent 2005). Note, however, that AB Aurigae is an object that is still surrounded by a substantial amount of non-disk-related circumstellar
material, which may contribute to the NIR flux.
The key to distinguishing these models from each other is to spatially resolve the NIR disk
A VLTI-PIONIER large program
B. Lazare↵ et al.: Few pu↵ed-up rims found in large interferometric survey of Herbig AeBe stars
Table 3: Photometric measurements used to determine the SED of our program objects. SAAO measurements are used by default.
WISE measurements are used only for plots but not for the SED fits. The Key field is a pointer from Table 4.
m
H
❖
❖
❖
❖
❖
❖
Reference
Monet et al. (2003) (USNO)
Zacharias et al. (2004) (NOMAD)
et al. (2000) (TYCHO)
4 Telescopes, 6v2 and 6Høg
closure
phases
Cutri
et al. (2003)
(2MASS)
Cutri & et al. (2012) (WISE)
Oudmaijer
al. (2001)
(EXPORT)
H band: 1.65 m, peak 1500
K atet 2.2
mic,
Malfait et al. (1998)
dust sublimation
Mendigutía et al. (2012)
Tannirkulam et al. (2008a)
SAAO 0.5m and 0.75m telescopes
PIONIER:
100 m baseline, 3mas, 1 AU @ 300 pc
Bands
20 R2 I
B2
BVR
15V
B
JHK
3.3,
10 4.6, 12, 22 µm
UBVRI
UBVJHKLM
J 5H K L M
UBVRIJHK
U0B V R I J H K
B0
A0
Key
1
2
3
4
5
6
7
8
Epoch
1974 - 2000
unknown
1989 - 1993
1998 - 2001
2010
1998 - 1999
1989 - 1992
1992 - ?
2004-2006
2014
F0
G0
of program objects (top) and
HD293782
mk )s are the intrinsic stellar colors derived from the Fig. 1: Statistics of H magnitudes
published (SIMBAD) spectral type and an interpolation in spectral types (bottom).
Usno
Tycho
the tables of Pecaut & Mamajek (2013);
2mass
– rk is the extinction ratio AV /E(mk mV ) from the extinction
Wise
Exp
−9
law of Cardelli et al. (1989) with R = 3.
a spectral
faintest objects
10 resolving power of R ⇡ 15, while theTanni
– (mV
30 AT Nights
SAMPLE:
SaVis2
The fit(Hillenbrand.92,
parameters are:Thé94, Malfait98)
AeBe
νFν (erg cm−2 s−1)
SaIr2
were observed in broad H-band (R ⇡ 5).
51 H
Data were reduced and calibrated with the pndrs package
– F⌫sV , the absorption-free stellar flux in V band
described in Le Bouquin et al. (2011). Each observation block
– F⌫dH , the absorption-free dust flux in H band
❖ 45 Non binary
– AV
(OB) provides five consecutive files within a few minutes. Each
– T dp , the blackbody temperature of the dust component
file contains six squared visibilities V2 and four phase closures
❖ 26 High AQuality
over the three spectral channels. Whenever possible,
derived parameter of interest is fd , the fraction of the observed dispersed
−10
10
flux at 1.63µm contributed by the dust component.
the five files were averaged together to reduce the final amount of
The
procedure
and
its
result
are
independent
of
the
distance
data to be analyzed and to increase the signal-to-noise ratio. The
❖ Additional Photometry
of the object and of the star’s luminosity class (except for a pos- statistical uncertainties typically range from 0.5 to 10 deg for the
sible small dependence of colors on luminosity class).
phase closures and from 2.5% to 20% for the squared visibilities,
❖ Litterature The
use of a standard(SAAO
value for Sutherland)
the extinction ratio R = 3
+ UBVRIJHK
0.5
1
2
3
5
10
depending 0.3
on target
brightness
and
atmospheric
conditions.
λ (µ)
might be objected, for the (unknown) part of the extinction that
Each observation sequence of one of our targets was immeis circumstellar. On the other hand, we are reluctant to derive
by the observation
of bands
a calibraFig. 2:preceded
Example and
of a followed
fit to photometric
data. Only the
more than four fit parameters with eight observed fluxes. To diately
UBVRIJHK
are
used
in
the
fit;
the
Wise
data
are
for
reference
tion
star
in
order
to
master
the
instrumental
and
atmospheric
regauge the impact of that assumption, we have repeated the fits
❖
have been r
typical calib
5% for the s
tive wavelen
the-sky orie
and is consi
2.3. Photom
Magnitudes
listed in Ta
objects, and
dating back
try of most
land 0.5 m
few norther
or infrared
measureme
relative to
two years) t
vious meas
error estima
and was us
VV Ser fro
measureme
The pho
listed in Tab
3. Data Pr
In this secti
from reduce
closures) to
below deals
maining su
data.
3.1. Model
For each ob
An example of HQ Pionier data
B. Lazare↵ et al.: Few pu↵ed-up rims found in large interferometric survey o
–5–
1
0.9
Extended flux
HD100453
0.8
0.7
Wagner et al. 2015
nd IFS images of HD 100453, verifying the detection of the spiral disk
Results
The spiral disk
n of a ring surrounded by two spiral arms that is clearly
scattered light, shown in Figures 1 & 2. Our observations
the star with fractional luminosity (f = Ldisk /L? ) of fY =
0.049, and fK2 = 0.103. The apparent furthest extent of
W and SE. Assuming a circular geometry, we calculate the
face on. The NE spiral intersects the ring between PA=22o
PA= 15o . The SW spiral intersects the inner ring between
extends to PA=155o . The apparent opening angles of the
⇠ 30o , respectively.
Resolved inner rim
0.5
fs ( 0 / )ks + Vc fc
V=
( fs + fh ) ( 0 / )ks +
0.4
0.3
.
v2
3.
0.6
c.d
1.
e
large distances from the
gion of the grains’ therm
component arises from
spectral index. We verifi
—same spectral index as
cumstellar component—
The complex visibility o
fied form in Eq. 2) can b
s
where 0 is a referen
sh
Unresolvedcentral
stellarPIONIER
flux
chann
of the three components
0.1
stellar, and halo respect
0
displays several finer structures, illustrated in the azimuthal
that in contrast with Eq
0
20
40
60
80
100
First, we identify sharp axisymmetric drops in brightness to
g ⇠0. 03 outside of the inner working angle for more than
Mλ
length dependence conta
the signature of a gap in the disk. It is important to note
Chromatic dispersion:
ity of the circumstellar c
s is outside of the actual coronagraphic mask, whose radius
6
is a↵ected by scattering and residual light, and hence the
Fig. 3: A plot of visibility
squared versus
B/ (inbetween
units of 10star/hot
). as concerns
Temperature
difference
dust both the fun
s the radius at which significant contrast may be obtained.
e are what would be expected of a gapped disk with a wall
Data points from the three wavelength channels of Pionier are cal values of the parame
nclination of ⇠34 .
identified by the shape and shading of the symbols; diamonds: Several functional forms
00
o
0.2
Table 8: Parameters for ring fits. Redundant parameter fs is also
included, see Table 6
kc
fc
fh
fLor
la
lkr
cos i
✓
A&A proofs: manuscript no. HAeBeLP_master
ck , sk
spectral index of circumstellar component
fractional flux of circumstellar component
fractional flux of halo component
weighting for radial profile
(log) half light semi major axis
(log) ratio (kernel radius) / (ring radius)
axis ratio of isophotes
position angle of major axis, East from North
cosine and sine amplitudes for mode k
– our goal is not to fineit represents reality; a
ments on some large-s
Parametric fitting: the method
HD 58647
5
1
0.8
10
0.8
0
0.6
0.4
−10
0.4
0.2
−20
0.2
0
−30
0
50
B (Mλ)
100
0
50
B (Mλ)
100
−5
Ring m=0
0
Ring m=1
−5
5
0
dAz (mas)
0
0
B (Mλ)
❖
1 star
❖
10
HD139614
RingM1
1
fully
resolved
component
1
−100
(k)
1
00
−20
100
−10
−5 10
0
0
5
−5
−20
−10 20
10
0
−10
−5
−20 5
20
5
0
0
0
0
−5
50
0
0.5
5
0
−10
−5 10
0.2
10
0.1
0.05
0.02
0
0
−5
−20
−10 20
50
B (Mλ)
0
−10
10
100
−5
−20 5
20
5
0
0
0
−5
5
0
0
−5
0
−10
−5
−20 5
5
0
0
50
−5
−10
−5 10
10
20
0
−5
0
dAz (mas)
−5
−50
0
0
dAz (mas)
−20
−10 20
5
0
−5
dAz (mas)
−20
−10 20
0
−50
−5
B10 (Mλ)
0
−10
−20 5
dAz (mas)
4
RingM1
Power
law with inner boundary
In some, if not all, current2 models for inner disk rims, one expects the brightness distribution to have a sharper cuto↵ inwards
0 we tested models with a radial disthan outwards. Accordingly,
tribution:
−2
(
0
if r < ar
F0 (r) = ar 3
(9)
r
if r ar−40
50 2⇡
100
50
100
1 ring of variable0 width (up to no ring
at all)
0.6
v2
φC (o)
v2
5
10
0
−5
dAz (mas)
5
HD142666
3.7.
0.8
0.6
0
Fig. 7: Examples of(l)
model fits, showing four objects and fit results at four levels of complexity of the fit model.
0.8
0.4
0.4
−5
❖
Ring m=2
5
0
0
5
5
❖
0
−5
−5
−100
100
−5
0
−50
10
1
dEl (mas)
0.1
0.05
0.02
dEl (mas)
0.2
B (Mλ)
dEl (mas)
0
0
−10
dEl (mas)
5
5
10
50
−5
B (Mλ)
5
50
20
−10
−5 10
0
5
1
0.5
5
5
100
5
0
4.1.1. Temperatures from
φC (o)
0
Ellipsoid
dEl (mas)
0.6
20
φC (o)
20
v2
1
10
RingM1
dEl (mas)
HD100546
B (Mλ)
30
RingM1
φC (o)
v2
HD100453
4.1. Dust temperatures
HD 98922
HD 45677
HD 100453
– we discuss our sampl
ties, which makes the
not implying that real
0.2
Azimuthal
modulation
allowed
to
reproduce
radiative transfer effects
0
−10
0
0.2
0
50
B (Mλ)
100
0
50
B (Mλ)
100
0
B (Mλ)
B (Mλ)
and similar models with an exponent di↵erent from 3, or an ex-
Figure 8 shows the histogr
to fd > 0.1 to exclude obje
meaningful. The histogram
above 2200K or below 12
of such outliers, we make
The median value o
3.22). The median of th
0.026, consistent
with an
⇣
median log10 (T dp ) med
dispersion being mostly fr
The median value for
sublimation temperatures
regarding disks around HA
(2010) and references ther
Königl (2012). However, i
exist over a range of tempe
ature; furthermore, our fit
bias. To address these tw
diative transfer code to ge
(SED) from a few simple
two spectral types (for the
The inner rim radius of ea
imum grain temperature w
tion temperature T sub for
silicates, and 1800K for c
discuss dust temperatures,
subsection 4.2. The values
K) are used as input to th
for deriving T dp from the
Fig. 9. The results are sum
T d,fit are plotted on Fig. 8;
to agree better⇣ with
⌘ the ob
ence in log10 T dp betwee
is 2.7 times larger than th
We conclude that:
Dust temperature
B. Lazare↵ et al.: Few pu↵ed-up rims found in large interferometric sur
18
❖
❖
❖
❖
16
14
12
In reality spread of Temperature
below T sub
10
8
−9
10
4
2
0
1000
Favours carbon grains (big?)
Additional unresolved component
not taken into account ?
−8
10
6
Fit on synthetic SED shows peak
temperature correspond to dust rim
Temperature of ~1800 K
Interferometric temperatures
systematically lower
10
Models
Tsub 1800K
ν Fν (erg cm−2 s−1)
❖
Temperature extracted from single BB
fit to SED: peak @ 1650 K
log10(Tdi)
❖
−7
Models
Tsub 1500K
−10
Star
StarS
StarT
DustT
DustS
DustT
Tot
ModS
ModD
ModT
10
A&A
proofs:2600
manuscript no. HAeBeLP_master
2200
Fig. 9: Sample fit of
FOST program. The
the
of sublimation
the MCFOSTtem
m
Fig. 8: Histogram of T dp (log scale) derived from fits to photo- grains.
sion
(dash-dot);
tota
3.4
metric data, restricted to non-binary objects having a fractional
Our IN05-like m
contribution of the dust to the H-band flux fd
0.1. Dashed ity between star and
dius:
1.2µm
o
are
used
forwhile
star an
lines: values of T dp obtained when fitting, with same procedure,
two
grain
radii:
0.35
flux
values
in
the
eig
models with sublimation temperature 1500K (e.g., silicate) or
distribution
of H-ba
star+dust model.
A
3.3
1800K (e.g. carbon); see text and table 9
appears originals,
to provideo
THM07
trum, even though t
Table 9: Properties of simple models and the respective fits to of dust and radiative
tures.
the dust temperature. Common properties as follows. Viewing with azimuthal sym
3.2
Variants of each
angle cos i = 0.55(i ⇡ 56 ); distance 122 pc; Rout = 125AU;
scale height 0.07 AU at r = 1AU; flaring exponent 1.10; dust types of central star
mass 0.65 10 3 M ; grain size: amin = 0.1 µ m, amax = 1000 µ m, each
bandmodel
photometry)
were adjf
exponent 3.5.
band interferometry
imum
grain temper
3.1
(see 2.1 above);
adjusted
⇢ / ` 1(iv)
to
A
total
of
35 objects
Model
T e↵
Lb
Rin
Grains T d,max T d,fit
ticipating the
obser
(K)
L
AU
K
K
intercepted
byranges
the d
First, the
A1s
9250
36
0.50 silicate 1520 1370
expected
grain sub
relative
thickness
z/
3
A1c
0.36 carbon 1880 1580
clear
be
A8
andcorrelation
A0 variants
B2s
20600 2280 4.5 silicate 1460 1330
ues of T dibased
are predo
variants,
on F
B2c 3
2.9
Remember
that T dpA
vided
in Appendix
3.1
3.2 carbon
3.3 1800 3.41637
HAeBe
object in to
th
log10(Tdp)
Additionally,
composition in the s
1400
1800
Tdp
The radius luminosity relation
B. Lazare↵ et al.: Few pu↵ed-up rims found in large interferometri
1
10
RadiusLuminosity RingM1.pdf
HD45677
HD190073
HD95881
0
10
(au)
HD37806
HD98922
MWC158
VV_SER
a
t
HD58647
AB_AUR
HD144668
HD100453
HD100546
HD163296
HD139614
HD144432
HD145718
HD97048
HD142527
HD150193
HD142666
HD158643
HD169142
HD259431
−1
10
HD85567
MWC297
HD143006
−2
10
•
•
•
0
1
2
log10(Lbol /L )
3
4
One possible reference radius-luminosity relation:
Tsub ~1800
K Luminosity-size plot for HQ objects. Filled symbols repFig. 21:
resent efficiency:
objects that1 meet
following criteria: (1) fc > 0.1, (2) specHigh cooling
(big grains)
tral type
A9 or earlier;
(3) luminosity class V or IV. Dashed
Backwarming
coefficient:
Cbw ~1
line represents the sublimation radius asub,1800 for T sub = 1800K,
RingM1
HD2594
40
1
0.8
20
0.8
The width of inner rim
0.6
0.4
0
v2
1
φC (o)
v2
HD45677
−20
0.2
−40
0
−60
0
50
B (Mλ)
100
0.6
0.4
w = ring radius / half light A&A
radius
0.2 manuscript
proofs:
w = 1 -> nor ring or ring not resolved
0
50
B (Mλ)
1
0
100
0.9
20
0
5
0.8
1
0.2
0.1
0.05
0.02
dEl (mas)
0
0.6
w
0.5
B (Mλ)
dEl (mas)
0.7
50
0.5
0
0.4
0
−50
0.3
0.2
−20
20
0
dAz (mas)
−20
0.1
0
50
0
−50
B (Mλ)
0
10
−5
5
1
10
a (mas)
Some objects do want rings(c)
(w <1.0)
preferfitfat
ones
Fig. 13:but
Parametric
results
for(w
the>0.5)
26 HQ objects. Normalized
ring fitting
width wprocess?
versus half flux radius a. See text.
is it a result of the
Article number, page 34 of 43page.43
0.3
1.2
0.2
1
0
The width of the inner rim
0
χ2r Ratio
0.1
χ2r(m=2) / χ2r(m=0)
1
10
10
a (mas)
0.8
0.6
0.4
Fig. 13: Parametric fit results for the 26 HQ objects. Normalized
ring
width w
versus
flux radius of
a. MCFOST
See text. models at
Method:
fitting
ofhalf
a population
in model;
inspired by Isella + Natta 2005
0.2
B. Lazare↵ et al.: Few pu↵ed-up rims found in large interferometric survey of Herbig A
different distances and inclinations (Pinte 2006)
w
0.6
0.5
0.4
1000
0.03
0
800
600
Fig. 15: Improvement
of 2 (or lack of-) when increas
400
number of azimuthal
modes
in the fit model, the 2 c
−5
0.01
200
ing both the terms
from v2 and cp. Note that, in contra
0
20.4 0
0.1
0.2
0.3
5
0ring model
−5
Fig.r 12,
the for the un-modulated
includes
(au)
α off (mas)
sure residuals, to be comparable45°
toinc;
the
other two fit mode
1.65µm
0.06
1800
max T,azimuthal
smallest to largest grains
modulation.
.
1600
0.05
0.3
0.2
1200
0.02
0.7
5
1400
0.04
z (au)
0.8
45° inc; 1.65µm
1800
1600
0.05
Pure ellipsoids
0.9
0
0.04
1768
1828
5
1400
1200
δ off (mas)
1
max T : 1824
HD259431
HD143006
HD85567
HD169142
HD158643
HD97048
HD95881
HD142666
δ off (mas) VV_SER
HD142527
HD144432
HD139614
HD145718
HD37806
HD58647
HD190073
HD150193
HD98922
HD144668
AB_AUR
MWC158
HD163296
MWC297
HD100546
HD100453
HD45677
0.06
z (AU)
1000to first order; (b) a↵ects the phase c
impact visibilities
0
Fig. 11. G
800
Conversely, an m=2
modulation (alone) a↵ects (above)
visibili
0.1
and t
0.02
600
files; in the c
not closures, to first order.
ature distribu
400
0
−5
0
1
onto 10
that for
0.01
The
values
of
the
three
models
listed
in
Table
ar
10
10
200
= 1.65µm
a (mas)
low brightnea
pared in Fig. 15, 0with the un-modulated model serving
0
5
0
−5
0.1
0.2
0.3
0.4
root s
erence.
ordered by square
increas
off (mas)
r (AU) As in Fig. 12, the objects αare
object
locate
Fig. 14: Similar to Fig. 13, but with fits of Mcfost models in, thm (value of a). One can see that most, but not all, of the
2 compare one-to
and ell added.
we
resolved
objects
show an improvement
ofcannot
when mod
1.1 model;
thm
inspired
by Tannirkulam
Harriesthough
Monnier
2007
the "real" width (if only because
is introduced; m = 1 modulationtion
seems
to make
di↵
for the latter),
this plotamake
1
of the LP objects are wider than t
more than m = 2.
LP (HQ)
in
thm
ell
0.03
The vertical structure
Results : vertical structure
First approach : fraction of stellar flux reprocessed to NIR
imax =
π
2
−θ
f NIR fbol =
Ω
= sin θ
4π
B. Lazare↵ et al.: Few pu↵ed-up rims found in large interferometric survey of Herbig AeBe starsA&A proofs: manuscript no. HAeBeLP_master
Statistics of inclination
Statistics of reprocesses flux
frep = LNIR,d / Lbol,∗
We can, however, compare the median value of the observed
11
0.4
8
sample with the similar metric for a model disk
to derive a typinon−Bin
non−Bin
non−Bin (Ellipsoid)
cal10value. The median values of frep are: for the 45 LP non-binary0.35
HQ
(Ring)
7
objects:
0.18; for the in_A0 Mcfost models: 0.19; for the thm_A0
9
models:
0.18.
median--for
Mcfoststatus
models
is restricted to in- 0.3
23
Gravity GTO
-- The
26-JAN-1016
HAeBeLP
report
6
8
clinations
for which the extinction of the central star AV < 3.
Remember
(section 4.2) that the thickness of the in and thm0.25
7
5
models was adjust to achieve such an agreement. Just because
6
we base our discussion on the median value does not mean there 0.2
4
is a 5unique value; indeed the scatter of frep as shown on Fig. 20
might
3
4 be real, reflecting the diversity of the objects include in0.15
our sample. Figure 19 (right-hand panel) provides a hint that one
3
2
of the
"explanatory variables" might be the spectral type: there 0.1
seems
2 to be a transition in the frep around spectral type A0.
1
There is in principle a flaw in this procedure. In the previous0.05
1
subsection we argued that the flaring of the outer disk restricts
0
0
the 0observed
HAeBe0.2disks to
the inclination
range 10.6 cos(i)
0
0.2
0.4
0.6
0.8
1
0
0.1
0.3
0.4
0.5
B0
B5
A0
A5
F0
F5
cos(i)
0.45, while we are now
the observed statistics of frep
frepcalibrating
= LNIR,d / Lbol,∗
Spectral type
against models without any outer flaring, that remain observable
Fig. 18: Histogram of fit values for cos(i). The histogram for theFig.over
widerHistogram
range of inclinations,
typically
cos(i)
0.8. luminosity reprocessed to near infrared for non-binary objects.
19:aLeft:
of the fraction
frep of 1stellar
bolometric
Observed
corresponds
z /model
r ~ 0.2
@
τ =the
1 model
Observed
statistics
corresponds
to type.
z / r ~ 0.44
26 objects ofstatistics
the HQ sub-sample,
fitted with thetoRing
isFour
However,
the strongest
dependence
of frepplot
veroutliers
are
outsidewith
the limits
of the plot.
Right:
Scatter
of frep versus spectral
superimposed over the histogram for the 45 non-binary objects, sus inclination —a vertical-wall rim— is known, even before
fitted with the Ellipsoid model. Parameter cos(i) is in fact robust: the present work, to be incompatible with the visibility curve,
its value is not sensitive to the particular fit model used.
and even the less extreme in model is excluded on the same where ✏ = Qabs (T sub )/Qabs (T ⇤ ), the cooling efficiency, is the ratio
0.5 see Fig. 14. Models such as thm that have a radially ex- of Planck-averaged absorption cross-section for the considered
basis;
LP non−binary
tended
emission
also have a comparatively weaker dependence grain species.
Table 11: List of three outlier objects excluded fom the his0.45
linear fit
inclination;
see Fig.20. This means that selection
The line drawn on Fig. 21 expresses Eq. 12 with T sub =
togram on the left panel of Fig. 19, and additional object is out- of frep versusmodel
in_A0
0.4 on cos(i)
e↵ects
dothm_A0
not a↵ect significantly our determination of 1800K, ✏ = 1 and Cbw = 1. The sublimation temperature is
model
side the bounds of the scatter plot in the right panel.
the thickness z/r of the inner disk.
Species
Astron. silicate
Olivine
Forsterite
Iron
Iron⋆
Corundum
Orthopyroxene⋆
Quartz⋆
Water ice⋆
Troilite⋆
Chemical
formula
MgFeSiO4
MgFeSiO4
Mg2 SiO4
Fe
Fe
Al2 O3
MgSiO3
SiO2
H2 O
FeS
T subl
[K]
1300
1300
1150
1400
1150
1500
...
...
150
680
A
(Eq. 1)
28030.
28030.
26091.
21542.
20686.
40720.
30478.
26335.
2827.7
155.91
B
(Eq. 1)
12.471
12.471
13.418
6.6715
9.1134
18.479
14.898
11.184
7.7205
–4.9516
Reference for A and B
Reference for opacity
Pollack et al. (1994, P94)
P94
Cameron & Fegley (1982, CF82)
P94
CF82
CF82
P94
Schick (1960); Lamy (1974)
P94
P94 (note: T subl = const.)
Draine & Lee (1984)
Dorschner et al. (1995)
Servoin & Piriou (1973)
P94
Wrap up: scientific results
⋆
❖
❖
Not used in the present work, given for completeness.
Sublimation curves of the main species used in this work are
shown in Fig. 2. At vapour densities of 10−12...−10 g/cm3 , olivine
sublimates
around
K, iron
1400 K and corundum at
Dust @inner edge
has atsub
~ 1300
1800
K ..at large
1500 K. Corundum must have an abundance <1% of the total
carbon grains dust mass and thus will necessarily have a much lower partial
density than olivine or iron.
Koike et al. (1995)
Kama et al. 2009
Few objects fully resolved
3.3. Backwarming
❖
❖
❖
❖
❖
❖
Radius-luminosity
correlation
confirmed
If a dust
grain cannot cool
into empty space in every direction,
its temperature increases due to backwarming. Geometrically,
the backwarming factor Cbw can be defined as the ratio of a full
Ring structure is4πfound
into50%
ofangle
object
solid angle
the solid
subtended by the empty sky
seen by the grain. For each photon emitted into the sky area obwhen resolved scured by nearby matter, a similar photon returns, hence the term
“backwarming”. An estimate of the temperature of a dust grain Fig. 1. A schematic of the inner few AU of a disc. The colour map indion the rim surface is obtained from the equation The Astrophysical Journal,
780:42
2014 January 1 from <300 K (dark) to >2000 K (white). At the
cates
the(9pp),
temperature,
Wide rings are favoured (consistent with
latest physical models
Tannirkulam
07, Kama
L⋆
4π
π
=
ϵσT d4 ,
2
Cbw
4πRrim16)
09, Turner 14 Flock
where the energy absorbed by a grain at a distance R
(2)
from a
rim
Thickness z/r ~0.2
rim
than
star ofnear
luminosity
L⋆ larger
is equated with
that~0.1
emitted by it, assuming
a cooling efficiency ϵ. Rigorously, Cbw is wavelength-dependent
expected (Vinkovic
and its 2014)
value can be obtained by solving a complex radiative
Turner et al.
optically thin radius, Cbw = 1, dust can begin to condense. The classical
rim, at Cbw = 4, is a maximally backwarmed dust wall. The actual rim,
i.e. τ = 1 surface, is between these limits and preceded by an optically
thin zone.
On the rim, Cbw > 4 may also be true for special locations.
One can imagine this for a dust grain sitting at the far end of a
tunnel which extends into the rim (see also Appendix C).
transfer and dust sublimation and condensation problem, which
is done in the Monte Carlo code we employ. However, as we
Evidence for substantial
flaring
3.4. Monte Carlo radiative transfer with MCMax
show later, the description in Eq. (2), while simple, is useful in
interpreting the results of our more rigorous models, as well as Radiative transfer through the complex dust geometries we wish
other
work on the subject.
Other
things being
constant, the rim to consider requires a flexible method in terms of density strucHalo component
correlates
with1/2
Herbig
class
Figure 5. Synthetic images of the central region of the dusty Herbig disk with (left to right) no magnetic support, magnetic support throughout, and a magnetically
location scales as Rrim ∝ Cbw .
tures and dust composition. In addition, the coupled nature of
supported bump. The field of view is 2.5 AU wide, and the system is inclined 60◦ from face-on. The star is shown to scale at the center of each panel’s left edge. The
blue, green, and red channels
in each
image
correspond to wavelengths
1.25, 1.6,
and 2.2 µm, i.e.,structure
J, H, and K bands,
respectively.
The same logarithmic intensity
the
dust
temperature
and the
hydrostatic
also
calls for
Cbw = 1 yields the temperature of a grain in an optically
thin
scale is used in all three panels.
anis available
iterative
solution.
environment, and when T subl is considered it gives
theversion
small(A color
of this figure
in the online
journal.) The required flexibility and speed can be
est possible distance from the star a passive grain may orbit obtained by using Monte Carlo radiative transfer.
regime, where
the gas code
molecules’
mean free path exceeds the
On has
the near-infrared
from
2 µm
shortward,
Weother
useside,
here
the
Monte
Carlo radiative
transfer
MCMax
for longer than its sublimation timescale. This limit
been bump’s
Turner et al. 2014