THE ASTROPHYSICAL JOURNAL, 547:1077È1089, 2001 February 1
( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.
SPECTRAL ENERGY DISTRIBUTIONS OF PASSIVE T TAURI AND HERBIG Ae DISKS:
GRAIN MINERALOGY, PARAMETER DEPENDENCES, AND COMPARISON
WITH INFRARED SPACE OBSERVAT ORY LWS OBSERVATIONS
E. I. CHIANG,1,2,3 M. K. JOUNG,3,4 M. J. CREECH-EAKMAN,5,6 C. QI,6 J. E. KESSLER,7
G. A. BLAKE,6,7 AND E. F. VAN DISHOECK8
Received 2000 August 1; accepted 2000 September 21
ABSTRACT
We improve upon the radiative, hydrostatic equilibrium models of passive circumstellar disks con-
structed by Chiang & Goldreich. New features include (1) an account for a range of particle sizes, (2)
employment of laboratory-based optical constants of representative grain materials, and (3) numerical
solution of the equations of radiative and hydrostatic equilibrium within the original two-layer (disk
surface plus disk interior) approximation. We systematically explore how the spectral energy distribution
(SED) of a face-on disk depends on grain size distributions, disk geometries and surface densities, and
stellar photospheric temperatures. Observed SEDs of three Herbig Ae and two T Tauri stars, including
spectra from the Long Wavelength Spectrometer (LWS) aboard the Infrared Space Observatory (ISO),
are Ðtted with our models. Silicate emission bands from optically thin, superheated disk surface layers
appear in nearly all systems. Water ice emission bands appear in LWS spectra of two of the coolest
stars. Infrared excesses in several sources are consistent with signiÐcant vertical settling of photospheric
grains. While this work furnishes further evidence that passive reprocessing of starlight by Ñared disks
adequately explains the origin of infrared-to-millimeter wavelength excesses of young stars, we emphasize
by explicit calculations how the SED alone does not provide sufficient information to constrain particle
sizes and disk masses uniquely.
Subject headings: accretion, accretion disks È circumstellar matter È radiative transfer È
stars: individual (MWC 480, HD 36112, CQ Tauri, LkCa 15, AA Tauri) È
stars: preÈmain-sequence
1. INTRODUCTION
The energetics of the outermost regions of isolated disks
surrounding T Tauri and Herbig Ae stars is dominated by
passive reprocessing of central starlight. While many pro-
tostellar disks are actively accreting (see, e.g., the review by
Calvet, Hartmann, & Strom 2000), e†ects of viscous dissi-
pation on disk spectra manifest themselves most strongly in
the immediate vicinities of the central stars, i.e., in the
steepest portions of their gravitational potential wells.
Simple scaling laws illustrate the relative importance of
external irradiation versus accretion luminosity. The local
viscous luminosity per unit disk area decreases as 1/a3,
where a is the stellocentric distance. By contrast, the Ñux of
central stellar radiation striking the disk drops more slowly
as (sin a)/a2, where a is the angle at which starlight grazes
the disk surface. Vertical hydrostatic equilibrium normally
1 Hubble Fellow.
2 Institute for Advanced Study, School of Natural Sciences, Einstein
Drive, Princeton, NJ 08540; chiang=ias.edu.
3 Theoretical Astrophysics, California Institute of Technology 130È33,
Pasadena, CA 91125.
4 Department of Astronomy, Columbia University, New York, NY
10027; moo=astro.columbia.edu.
5 Jet Propulsion Lab, MS 171È113, Pasadena, CA 91109 ; mce=
huey.jpl.nasa.gov.
6 Division of Geological and Planetary Sciences, California Institute of
Technology 150È21, Pasadena, CA 91125; qch=gps.caltech.edu, gab=
gps.caltech.edu.
7 Division of Chemistry and Chemical Engineering, California Institute
of Technology, Pasadena, CA 91125; kessler=gps.caltech.edu.
8 Sterrewacht Leiden, P.O. Box 9513, 2300 RA, Leiden, Netherlands;
ewine=strw.leidenuniv.nl.
ensures that disks Ñare outward such that a is a slowly
increasing function of a for where is the stellara ? R
*
, R
*
radius (see, e.g., Kenyon & Hartmann 1987). Hence, there is
always a disk radius outside of which the energy from stellar
illumination outweighs that of midplane accretion; in the
extreme case that the central star derives its luminosity
wholly from accretion, this transition radius is roughly 1
AU. The spectral energy distributions (SEDs) of young star/
disk systems longward of D10 km should thus closely
approximate those of passively heated disks, even when acc-
retion is ongoing.
Hydrostatic, radiative equilibrium models of passive T
Tauri disks are derived by Chiang & Goldreich (1997, here-
after CG97). The passive disk divides naturally into two
regions: a surface layer that contains dust grains directly
exposed to central starlight, and a cooler interior that is
encased and di†usively heated by the surface (Calvet et al.
1991; Malbet & Bertout 1991; CG97; DÏAlessio et al. 1998).
CG97 compute SEDs of passive disks viewed face-on and
employ their model to satisfactorily Ðt the Ñattish infrared
excess and millimeter wavelength emission of the T Tauri
star GM Aur. The optically thin, superheated surface layer
is shown to be the natural seat of silicate emission lines (see
also Calvet et al. 1992).
In a second paper, Chiang & Goldreich (1999, hereafter
CG99) compute SEDs of passive T Tauri disks viewed at
arbitrary inclinations. They point out that the spectrum of a
nearly edge-on disk is that of a class I source for which lF
l
rises from 2 to 10 km (Lada & Wilking 1984 ; Lada 1987). In
general, class I sources are best described by a combination
of an inclined, passively heated disk and a dusty bipolar
outÑow or partially evacuated envelope. The fraction of
1077
1078 CHIANG ET AL. Vol. 547
class I T Tauri spectra that represent limiting cases of
simple, isolated, inclined disks is small (DÏAlessio et al.
1999) but nonzero (CG99).
This third paper in our series on passive protostellar
disks extends our work in three directions :
1. We reÐne equilibrium, two-layer models of passive
disks by (1) accounting for a range of particle sizes, (2)
employing laboratory-based optical constants of a suite of
circumstellar grain materials, and (3) solving numerically
the equations of radiative and hydrostatic equilibrium
within our original two-layer approximation.
2. We systematically explore how the SED of a face-on
disk depends on grain size distributions, disk geometries
and surface densities, and stellar photospheric tempera-
tures. Physical explanations are provided for all observed
behaviors of the SED.
3. We employ our reÐned face-on models to Ðt observed
SEDs of 3 Herbig Ae (HAe) and 2 T Tauri stars. These
observations include new spectra between 43 and 195 km
from the Long Wavelength Spectrometer (LWS) aboard the
Infrared Space Observatory (ISO) (Creech-Eakman et al.
2000, in preparation). The uniqueness of our Ðtted values of
disk parameters is assessed, and evidence for emission lines
from superheated silicates and ices is reviewed.
The input parameters and basic equations governing our
reÐned standard model are detailed in ° 2. Results, including
a systematic exploration of how the SED varies in input
parameter space, are presented in ° 3. Model Ðts to obser-
vations are supplied and critically examined in ° 4. There we
also compare our results to recent modeling e†orts by
Miroshnichenko et al. (1999). Finally, we summarize our
Ðndings in ° 5.
2. REFINED MODEL
2.1. Input Parameters
Table 1 lists the input parameters of our reÐned model.
Figure 1 exhibits schematically the zones of varying grain
composition in both the disk surface and disk interior. For
FIG. 1.ÈSchematic of zones of grain compositions (iron ] amorphous
olivine ] amorphous olivine mantled with water ice) for both the disk
surface and interior. The numerous vertical lines drawn in the surface
indicate that there we account for how grains of di†erent sizes sublimate at
di†erent stellocentric distances. Dashed lines divide the superheated
surface layers from the disk interior. Dotted lines mark locations of con-
densation boundaries for the reÐned standard model only.
ease of computation, and for want of hard observational
constraints on the detailed compositions and optical
properties of circumstellar grains, we limit ourselves to con-
sidering metallic iron (Fe, bulk density \ 7.87 g cm~3),
amorphous olivine bulk density \ 3.71 g(MgFeSiO
4
,
cm~3), and water ice bulk density \ 1gcm~3).(H
2
O,
These cosmically abundant materials span a wide range in
condensation temperature (and therefore stellocentric
distance), and in the cases of silicates and water ice, their
existence is conÐrmed by spectroscopic observations (see,
e.g., ° 4.3 of this paper). In reality, protostellar disks contain
many more kinds of solid-state materials than we have
incorporated. We have experimented with including addi-
tional grain compositions (e.g., graphite, organics, and
troilite), but in no instance do we Ðnd our conclusions
TABLE 1
INPUT PARAMETERS OF REFINED MODEL
Symbol Meaning Standard Value
M
*
........ stellar mass 0.5 M
_
R
*
......... stellar radius 2.5 R
_
T
*
.......... stellar e†ective temperature 4000 K
&
0
......... surface density at 1 AU 103 gcm~2
p ........... [d log &/d log a 1.5
a
o
.......... outer disk radius 8600 R
*
\ 100 AU
H/h ........ visible photospheric height/gas scale height 4.0
q
i
.......... [d log N/d log r in interior 3.5
q
s
.......... [d log N/d log r in surface 3.5
r
max,i
....... maximum grain radius in interior 1000 km
r
max,s
...... maximum grain radius in surface 1 km
T
sub
iron ....... iron sublimation temperature 2000 K
T
sub
sil ....... silicate sublimation temperature 1500 K
T
sub
ice ....... H
2
O ice sublimation temperature 150 K
M
DISK
a ...... total disk mass (gas ] dust) 0.014 M
_
a The total disk mass is not an explicitly inputted parameter but is derived from &
0
,
p, and the inner disk cut-o† radius,a
o
, a
i
\ 2 R
*
.
No. 2, 2001 PASSIVE T TAURI AND HAE DISKS 1079
changed qualitatively. Our goal is not to be slavishly realis-
tic but rather to highlight chief physical e†ects.
Thus, where local dust temperatures (\gas temperatures)
fall below K, the grains are taken to be spheresT
sub
ice B 150
of amorphous olivine mantled with water ice. For simpli-
city, the thickness of the water ice mantle, *r, relative to the
radius of the olivine core, r, is held constant. Where local
temperatures fall between and K, only theT
sub
ice T
sub
sil B 1500
pure olivine cores are assumed to remain. In innermost disk
regions where local temperatures fall between andT
sub
sil
K, the grains are taken to be spheres of metal-T
sub
iron B 2000
lic iron.
The iron or silicate cores in the disk surface (interior)
possess a power-law distribution of radii r between andr
min
r
max,s
(r
max,i
):
dN P r~q
s(i) dr , (1)
where dN is the number density of grains having radii
between r and r ] dr. Variables subscripted with ““ s ÏÏ
denote quantities evaluated in the disk surface, while those
subscripted with ““ i ÏÏ denote quantities evaluated in the disk
interior. Our standard value of places most ofq
i
\ q
s
\ 3.5
the geometric surface area in the smallest grains and most of
the mass in the largest grains. In practice, is Ðxed atr
min
10~2 km, while and are free to vary.r
max,i
, r
max,s
, q
i
, q
s
Generally since large grains tend to settler
max,s
\ r
max,i
quickly out of tenuous surface layers (see ° 3.3 of CG97). All
of the cosmically abundant iron is assumed to be locked
within grains. Following Pollack et al. (1994), we take 50%
of the cosmically abundant oxygen to be locked in ice.H
2
O
Values for all cosmic abundances are obtained from Allen
(2000), except for the abundance of oxygen, which is taken
from Meyer, Jura, & Cardelli (1998). Together, these
assumptions yield a fractional thickness, *r/r, for the water
ice mantle equal to 0.4.
Optical constants for amorphous olivine are obtained
from the University of Jena Database (http://
www.astro.uni-jena.de; see also et al. 1994). Long-Ja
ger
ward of 500 km, where optical data for silicates are not
available, the complex refractive index (n ] ik) for glassy
olivine is extrapolated such that n(j º 500 km) \ n(500 km)
and k(j º 500 km) \ k(500 km)(j/500 km)~1. Optical con-
stants for pure crystalline ice are taken from theH
2
O
NASA ftp site (ftp:climate.gsfc.nasa.gov/pub/wiscombe/
Refrac
–
Index/ICE; see also Warren 1984). Though employ-
ing the constants for a cosmic mixture of amorphous ices
see Hudgins et al. 1993) would(H
2
O:CH
3
OH:CO:NH
3
;
be more appropriate, we nonetheless adopt the data for
pure ice because the latter are available over all wave-H
2
O
lengths of interest, from the ultraviolet to the radio, whereas
the former are not. One consequence of using the constants
for crystalline (213È272 K) water ice as opposed to amorp-
hous (D100 K) ice is that spectral features due to trans-
lational lattice modes at 45 and 62 km are slightly
underestimated in width and overestimated in amplitude
(see, e.g., Hudgins et al. 1993). Optical constants for metallic
Fe are obtained from Pollack et al. (1994).
The inner cuto† radius of the disk, is Ðxed at 2 Fora
i
, R
*
.
T Tauri stellar parameters, this radius coincides with the
distance at which iron grains in the surface layer attain their
sublimation temperature. For the hotter HAe stars, the iron
condensation boundary occurs at a B 14È30 Inside theR
*
.
iron condensation radius, the disk may still be optically
thick to stellar radiation even if dust is absent. Opacity
FIG. 2.ÈEmissivities of ice-silicate / amorphous olivine), silicate(H
2
O
(amorphous olivine), and iron grains having three representative core sizes.
The thickness of the water ice mantle relative to the radius of the olivine
core is *r/r \ 0.4. Resonant features include the O-H stretching (3.1, 4.5
km) and H-O-H bending (6.1 km) modes in water ice; the Si-O stretching
(10 km) and O-Si-O bending (18 km) modes in silicates ; and the intermo-
lecular translational (45, 62, and 154 km) modes in water ice. Oscillatory
behavior near the onset of the Rayleigh limit (2nr/jB1) reÑects ““ ripple
structure ÏÏ arising from our use of perfectly spherical particles.
sources include pressure-broadened molecular lines and
Rayleigh scattering o† hydrogen atoms (see the appendix of
Bell & Lin [1994]). For simplicity, when modeling HAe
stars, we employ a one-layer blackbody disk that extends
from the iron condensation boundary to a
i
\ 2 R
*
.
2.2. Grain Absorption Efficiencies and Opacities
The grain emissivity, e(r, j), is equal to its absorption
efficiency and is calculated using theoryMie-Gu
ttler
(Bohren & Hu†man 1983; see their subroutine
BHCOAT.F).9 Figure 2 displays absorption efficiencies for
our ice-silicate, silicate, and iron spheres having three repre-
sentative sizes. The emissivity index in the Rayleigh limit,
b 4 d ln e/d ln l, equals 1.64, 2.00, and 0.50 for the three
compositions, respectively.
Well-known resonances at km include the O-Hj [ 20
stretching (3.1, 4.5 km) and H-O-H bending (6.1 km) modes
in water, and the Si-O stretching (10 km) and O-Si-O
bending (18 km) modes in silicates.
In crystalline water ice (and crystalline silicates), optically
active modes of vibration longward of D10 km are
““ intermolecular translational ÏÏ or ““ intermolecular rota-
tional.ÏÏ These involve collective movement of a molecule or
a unit cell with respect to other molecules/unit cells in the
lattice. The strengths, positions, and widths of these modes
9 Our grain emissivity, e, equals in the notation of Bohren &Q
abs
Hu†man (1983).
1080 CHIANG ET AL. Vol. 547
are more sensitive to the presence of chemical impurities
and to long range order in the solid (i.e., its degree of crys-
tallinity, or ““ allotropic state ÏÏ) than those of fundamental
stretching and bending modes at shorter wavelengths. In
principle, these intermolecular modes provide information
on the annealing history of initially amorphous, interstellar
material in the relatively high density and high temperature
environments of circumstellar disks. The intermolecular
translational modes in water ice evident in Figure 2 are
located at 45, 62, and 154 km (Bohren & Hu†man 1983, p.
278; Bertie, & Whalley 1969, Figs. 4 and 11). All ofLabbe
,
these features are positioned within the wavelength range of
the Long Wavelength Spectrometer aboard ISO. Note also
the blending of the 12 km intermolecular rotational band in
water ice (Bohren & Hu†man 1983) with the Si-O silicate
stretching mode at 10 km.
Oscillatory behavior in Figure 2 near the onset of the
Rayleigh limit (2nr/jB1) reÑects so-called ““ ripple
structure ÏÏ that arises from our use of perfectly spherical
particles (Bohren & Hu†man 1983) ; we expect real-world
deviations from sphericity to smooth out this artiÐcial
behavior.
In the disk interior, the opacity is given by
i
i
(j) \
n
o
t
P
rmin
rmax,i dN
dr
r2e(j, r) dr , (2)
where is the total density of gas and dust. Figure 3 dis-o
t
plays for our distribution of ice-silicate and silicate par-i
i
ticles in cosmic abundance gas.
2.3. Basic Equations
In the surface layer, dust grains are directly exposed to
stellar radiation. The grains attain an equilibrium tem-
FIG. 3.ÈMass absorption coefficients for our standard model distribu-
tions of ice-silicate and silicate particles in solar abundance gas. The same
resonant features found in Fig. 2 are seen here. We have smoothed these
curves to suppress so-called ““ ripple structure ÏÏ at km that arisesj Z 1000
from the sphericity of our particles having r B 1000 km. Note that these
curves are sensitive to our chosen andq
i
r
max,i
.
perature
T
ds
(a, r) \ T
*
A
R
*
a
B
1@2
C
/
4
Se(r)T
T
*
Se(r)T
Tds
D
1@4
, (3)
where is the emissivity of a grain of radius r aver-Se(r)T
T
aged over the Planck function at temperature T , and
/ B 1/2 is the fraction of the stellar hemisphere that is seen
by the grain.
In our two-layer formalism, exactly half of the radiation
reprocessed by the surface layer escapes directly into space.
The remaining half is directed towards the disk interior. Of
this latter half, a fraction, is absorbed by the1 [ e~&Wi
iXs,
disk interior, where & is the disk surface density and isSi
i
T
s
the opacity of the disk interior averaged over the spectrum
of radiation from the surface.10 In radiative balance,
/
2
(1 [ e~&Wi
iXs)
A
R
*
a
B
2
T
*
4 sin a \ (1 [ e~&WiiXi)T
i
4 , (4)
where is the temperature of the disk interior and isT
i
Si
i
T
i
the interior opacity averaged over the spectrum of radiation
from the disk interior.11 Note that is the common tem-T
i
perature of interior grains of all sizes which are assumed to
have thermally equilibrated with one another. Here a is the
angle at which stellar radiation strikes the surface :
aBarctan
A
d ln H
d ln a
H
a
B
[ arctan
H
a
] arcsin
A
4
3n
R
*
a
B
(5)
(cf. eq. [5] of CG97). The height of the disk photosphere, H,
is assumed to be proportional to the vertical gas scale
height, h, with a Ðxed constant of proportionality equal to 4
for our standard model. In reality, when dust and gas are
well-mixed in interstellar proportions, the ratio H/h
decreases slowly from D5at1AUtoD4 at 100 AU. In
modeling observed SEDs, we will allow H/h to be a Ðtted
constant parameter. In hydrostatic equilibrium,
H
a
\
H
h
h
a
\
H
h
S
T
i
T
c
S
a
R
*
,
(6)
where and g.T
c
4 GM
*
k
g
/kR
*
k
g
\ 3 ] 10~24
Equations (4) and (6) are two equations for the two
unknown functions, H(a) and Substitution of (6) intoT
i
(a).
(4) yields an algebraic equation for and the slowly varyingT
i
Ñaring index, c 4 d ln H/d ln a. We solve this equation for
and c(a) numerically on a logarithmic grid in stellocen-T
i
(a)
tric distance. Our procedure is described in detail in the
Appendix.
The SED of the disk equals the sum of emission from the
disk interior,
4nd2jF
j,i
\ 8n2j
P
ai
ao
B
j
(T
i
)(1 [ e~&ii)ada , (7)
10 In practice, we perform this average over a Planck function evaluated
at the temperature of the most luminous grains in the surface. For our
assumed size distribution, these dominant grains typically have radii
r B 0.5 km ; surface grains having r B 0.2È1 km are responsible for absorb-
ing D50% of the incident stellar radiation.
11 We perform this average over a Planck function evaluated at tem-
perature T
i
.
No. 2, 2001 PASSIVE T TAURI AND HAE DISKS 1081
and from the surface layers (above and below the disk
midplane),
4nd2jF
j,s
\ 8n2j(1 ] e~&i
i)
P
ai
ao
S
j
(1 [ e~qs)ada , (8)
where d is the distance to the source. The source function in
the surface, is the Planck function averaged over theS
j
,
ensemble of e†ective grain cross-sections in the surface
layer:
S
j
\
2/
r
min
rmax,s B
j
(T
ds
)(dN/dr)r2e(j, r) dr
/
r
min
rmax,s (dN/dr)r2e(j, r) dr
. (9)
In equation (9), the factor of 2 is inserted so that exactly half
of the incident radiation is reprocessed by the surface layer.
The normal optical depth of the surface layer, is similarlyq
s
,
averaged:
q
s
(j, a) \
/
r
min
rmax,s (dN/dr)r2e(j, r) dr
/
r
min
rmax,s (dN/dr)r2Se(r)T
T*
dr
sin a . (10)
3. RESULTS
3.1. Flaring Index
Figure 4 displays the behavior of the Ñaring index,
c 4 d ln H/d ln a. We conceptually divide the disk into three
annular regions, as was done in CG97 (see their ° 2.3.2). In
FIG. 4.ÈFlaring index, c 4 d ln H/d ln a, for our reÐned standard
model. As was done in CG97 (see their ° 2.3.2), we divide the disk into three
annular regions depending on the optical depth of the disk interior. In
region I, the interior behaves as a blackbody ; c increases from its Ñat disk
value of 1.125 B 9/8 to its asymptotic Ñared value of 1.275 B 9/7 as the
disk thickness becomes increasingly larger than the stellar radius. In region
II, the disk interior becomes optically thin to its own reprocessed radi-
ation; c increases as interior grains enhance their temperatures to compen-
sate for the inefficiency with which they reradiate. In region III, the interior
is transparent to radiation from the surface and cools quickly with increas-
ing distance, causing c to decrease.
the region marked ““ I,ÏÏ the disk interior is opaque to both
its own reprocessed radiation and to radiation from the
surface. Here c increases from its Ñat disk value of
1.125 B 9/8 to its asymptotic value of 1.275 B 9/7 as the
Ðrst two terms on the right-hand side of equation (5) grad-
ually dominate the last term. In region II, the disk interior
remains opaque to radiation from the surface, but is opti-
cally thin to its own reprocessed radiation. Here c steeply
rises with a because grains in the disk interior equilibrate at
relatively high temperatures to compensate for the relative
inefficiency with which they reradiate the incident energy.
Finally, in region III, the interior is transparent to radiation
from the surface (i.e., the inability of the inte-&Si
i
T
s
[ 1) ;
rior to absorb the incident energy causes c to decrease.
3.2. Disk T emperatures
Figure 5 exhibits temperature proÐles for the surface (T
ds
)
and for the interior The temperatures of grains in the(T
i
).
surface layer vary slightly with their sizes; in Figure 5, we
have chosen to plot for the size bin containing the mostT
ds
luminous grains, i.e., the logarithmic size interval that
absorbs the greatest fraction of incident stellar radiation.
For our choices of grain composition and size distribution,
these dominantly absorbing grains have radii r B 0.1È0.7
km. For reference, we also overlay in Figure 5 the tem-
perature of an imaginary blackbody sphere, which isT
BB
,
naked before half of the stellar hemisphere.
These temperature proÐles are largely similar to those
found in the simpler model presented in CG97 and help to
justify the approximations made there. At a given distance,
FIG. 5.ÈTemperature proÐles for the surface and for the interior in our
reÐned standard model. The temperatures of grains in the surface layer
depend on their sizes; here, the curve marked represents the size bin,T
ds
r B 0.5 km, containing the most luminous grains. The discontinuity in T
ds
at a B 6 AU marks the water condensation boundary in the surface,
outside of which ice coats silicate cores; the discontinuity in atH
2
O T
ds
a B 0.05 AU marks the silicate condensation boundary in the surface. For
reference, the temperature of a spherical blackbody, which is naked before
half of the stellar hemisphere, is shown as a dashed line.
![FIG. 1.ÈSchematic of zones of grain compositions (iron ] amorphous olivine] amorphous olivine mantled with water ice) for both the disk surface and interior. The numerous vertical lines drawn in the surface indicate that there we account for how grains of di†erent sizes sublimate at di†erent stellocentric distances. Dashed lines divide the superheated surface layers from the disk interior. Dotted lines mark locations of condensation boundaries for the reÐned standard model only.](/figures/fig-1-eschematic-of-zones-of-grain-compositions-iron-102b4v94.webp)
