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THE ASTROPHYSICAL JOURNAL, 511 : 896È903, 1999 February 1
( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.
ON THE THERMAL STABILITY OF IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS
PAOLA DÏALESSIO,1 JORGE CANTO ,1 LEE HARTMANN,2 NURIA CALVET,2 AND SUSANA LIZANO1
Received 1998 April 16 ; accepted 1998 September 3
ABSTRACT
The dusty disks of many young stars are probably heated mostly by absorption of light from the
central star. This stellar irradiation can control the vertical structure of the disk, particularly in the outer
regions. Because the irradiation heating is sensitive to the disk structureÈthe disk vertical thickness and
the tilt of the disk photosphere relative to the starÈthe possibility of an unstable feedback is present. To
study this problem, we present calculations of the evolution of perturbations in vertically isothermal
disks. We Ðnd that such disks are generally stable. In outer disk regions of T Tauri stars, linear analysis
indicates that the radiative cooling time is so short that temperature perturbations will be damped faster
than the disk structure can respond. Using our results for steady ““ alpha ÏÏ viscosity disks, we estimate
that this is true for distances larger than 2 AU (M0 /10~8 M yr~1)7@9(a/0.01)~7@9 for typical T Tauri
_
stars. Inside this radius, if the disk surface tilt (““ Ñaring ÏÏ) is still signiÐcant, numerical Ðnite-amplitude
calculations show that temperature perturbations will travel inward as they damp. We Ðnd that disk
self-shadowing has a small e†ect on the results because the perturbation is damped on a timescale
shorter than the time in which the shadowed disk region can respond. Our results help justify steady,
smooth treatments of the e†ects of irradiation on the disks of young stellar objects.
Subject headings : accretion, accretion disks È circumstellar matter È instabilities È
stars : preÈmain-sequence
1.
INTRODUCTION
the heating produced by irradiation exceeds that resulting
from viscous dissipation of accretion energy near the disk
midplane, the outer layers of the disk would be hotter than
the interior, owing to nongray opacities and to the oblique
entry of starlight into the disk (see also Malbet & Bertout
1991). Irradiation-produced temperature inversions may
explain the 2.2 km CO overtone band emission seen in some
objects (Calvet et al. 1991), as well as silicate emission features in the mid-infrared range (Calvet & Hartmann 1992 ;
Chiang & Goldreich 1997).
Since irradiation heating can play such an important role
in determining the vertical structure of T Tauri disks, it is
important to investigate the stability of such disks to ripples
in their surfaces. Tilting the disk photosphere away from the
disk plane, toward the central star, increases the irradiation
heating, which increases the disk temperature. In turn, this
increases the local disk scale height, which might yield an
even larger tilt with the consequent instability.
In this paper we consider the problem of the stability of
nonaccreting, irradiated disks, approximating the disk as
having a well-deÐned photospheric level where it absorbs
the central starÏs light and assuming vertical isothermality.
We show that such disks are stable to rippling perturbations of their surfaces and argue that this result probably
applies to realistic protostellar disks. In ° 2 we consider
small perturbations adopting disk parameters typical of our
detailed structure calculations for T Tauri disks (DÏAlessio
et al. 1998). In ° 3 we consider the problem in more generality, which allows us to account for the di†ering slope of
the surface in inner and outer disk regions. In ° 4 we consider Ðnite-amplitude perturbations, and in ° 5 we present a
discussion of the results.
It has become increasingly clear that dusty preÈmainsequence disks are signiÐcantly heated by absorption of
light from the central T Tauri star (““ irradiation ÏÏ), and that
this heating plays an important role in establishing the
physical disk structure. The Ðrst clue to this came from
observations of the infrared spectral energy distributions of
T Tauri disks (Rydgren & Zak 1987), which implied disk
surface temperatures T that declined with increasing radius
R more slowly than the T P R~3@4 predicted by simple
optically thick, Ñat disk models, whether powered by steady
accretion (Lynden-Bell & Pringle 1974) or irradiation
(Adams , Lada, & Shu 1987). Kenyon & Hartmann (1987)
showed that the required disk-temperature distributions
could be explained by irradiation if preÈmain-sequence
disks are ““ Ñared,ÏÏ i.e., their surfaces curve away from the
midplane, which allows the disk to absorb more light from
the central star. Since this Ñaring increases the local disk
temperature, it increases the local disk scale height ; in outer
disk regions, irradiation will dominate disk heating, and
thus a self-consistent determination of disk structure
requires the simultaneous solution of irradiation and vertical hydrostatic equilibrium (DÏAlessio et al. 1998). The
Ñared disk model has been conÐrmed by Hubble Space
T elescope images of the young stellar object HH 30
(Burrows et al. 1996), which clearly show the curved disk
surfaces in scattered light.
A second clue to the importance of disk irradiation was
provided by observations of emission features in disk
spectra (Carr 1989 ; Cohen & Witteborn 1985). Calvet et al.
(1991) and Calvet & Hartmann (1992) pointed out that if
2.
1 Instituto de Astronom• a, UNAM, Ap. Postal 70-264, Cd. Universitaria, 04510 Mexico D.F., Mexico ; dalessio=astroscu.unam.mx,
lizano=astrosmo.unam.mx.
2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138 ; hartmann=cfa.harvard.edu,
ncalvet=
cfa.harvard.edu.
DISK SURFACE STABILITY : OUTER DISK
We begin with an analysis modeled after the work by
Cunningham (1976), who investigated the problem of stability of disks around black holes. We assume that the disk is
vertically isothermal and optically thick to the stellar irra896
IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS
diation. We further assume that the disk has a well-deÐned
photosphere (Kenyon & Hartmann 1987), i.e., that the
stellar photons are absorbed in a relatively narrow layer in
the disk. Finally, we assume that the disk vertical structure
responds instantaneously to a perturbation in the irradiated
Ñux (caused by changes in the position of the disk
photosphere).
To Ðx ideas in an appropriate context, we Ðrst consider
small perturbations and adopt speciÐc results for the background state of the disk motivated by our detailed structure
results for typical disk models in their outer regions, where
Ñaring is important (e.g., DÏAlessio et al. 1998).
The irradiation Ñux is given by
L
k
F \ * k \ pT 4
,
irr 4nR2
* r2
(1)
where L is the stellar luminosity, T is the stellar e†ective
*
*
temperature, r is the cylindrical radial coordinate R in units
of the stellar radius R , and k is the cosine of the mean
* stellar radiation and the normal
angle between the incident
to the disk photosphere. For the outer, Ñared disk regions,
the last quantity can be approximated by
A
B A
B
h d ln h
1 dh h
[ \
[1 ,
k^
2r d ln r
2 dr r
h \ h T 1@2r3@2 .
(3)
1
The unperturbed, steady state, disk temperature T (r) is
given by the balance between heating by irradiation0 and
radiative cooling, given by
pT 4 \ F
,
(4)
0
irr,0
where the subindex 0 corresponds to the unperturbed quantities.
Equation (4), along with equations (1), (2), and (3), is satisÐed by a power-law temperature distribution,
T (r) \ T r~3@7 .
(5)
0
1
Here T and h are scaling coefficients determined by the
1 disk 1parameters.
particular
Now we perturb the temperature of the disk, maintaining
vertical isothermality and assuming the perturbation is
small and axisymmetric. We take
T \ T (r)(1 ] v) ,
(6)
0
where v > 1. The perturbation in temperature introduces an
imbalance between heating and cooling, which results in
evolution of the perturbation, described by
LT
\ F [ pT 4 ,
irr
Lt
C AB D
M0 )
1 1@2
1[
,
(8)
3nac2
r
s
where ) is the Keplerian angular velocity ) \ (GM /R3)1@2,
*
M is the stellar mass, G is the gravitational constant, a is
*
the viscosity parameter (Shakura & Sunyaev 1973), and c is
s
the sound speed. We assume that the timescale for the perturbation evolution is smaller than the viscous timescale,
and that the speciÐc heat is constant in time. For a powerlaw (unperturbed) temperature radial distribution T (r) \
0
T r~q and for r ? 1, the speciÐc heat C(r) \ k&/m is also a
1
power law in radius and can be written as C(r) \ C rq~3@2.
1
Here k is the Boltzmann constant, m is the mean molecular
mass, and the constant C is given by
1
GM 1@2
M0
*
.
(9)
C \
1 3naT
R3
*
1
Assuming that the disk is always in hydrostatic equilibrium, the temperature perturbation results in a perturbation in the gas scale height,
&\
A B
A B
v
,
h \ h r3@2T 1@2(1 ] v)1@2 B h r3@2T 1@2 1 ]
1
0
1
0
2
(2)
where we are assuming k > 1. The height of the photosphere h is taken to be proportional to the gas density scale
height (Kenyon & Hartmann 1987 ; Miyake & Nakagawa
1995). Thus, the photospheric height (in units of stellar
radii) can be written as
C(r)
(Pringle 1981)
897
and
A
B
2 r Lv
d ln h
.
[1 \ ]
7 2 Lr
d ln r
(10)
(11)
Since the speciÐc heat is constant in time, we have
LT
Lv
B C(r)T
.
(12)
0 Lt
Lt
Substituting equations (10) and (11) into equation (1) and
using the thermal equilibrium equation (4), one obtains that
the irradiation Ñux is given by
C(r)
A BA
B
2 r Lv
]
.
7 2 Lr
On the other hand, the radiative cooling is
v
7
F \ pT 4 1 ]
irr 2
0
2
(13)
pT 4 B pT 4(1 ] 4v) .
(14)
0
Substituting equations (12), (13), and (14) into equation
(7), the evolution of the perturbation is given by
4 C(r) Lv
Lv
] 2v [ r \ 0 ,
(15)
7 pT 3 Lt
Lr
0
where we have retained only terms linear in the perturbation.
Following Cunningham (1976), we search for a solution
of the form
v(r, t) \ r2w(t, f) ,
(16)
Lv
Lw
\ r2
,
Lt
Lt
(17)
Lv
Lw
\ 2rw ] r2
bbrb~1 .
Lr
Lf
(18)
where f \ brb. Then
(7)
where we are neglecting the radial transport of energy.
In equation (7), C(r) is the speciÐc heat per surface unit of
the disk, which is proportional to the surface mass density.
For a standard a viscous disk with a constant mass accretion rate M0 , the mass surface density can be written as
and
898
DÏALESSIO ET AL.
& Nakagawa 1995 ; Canto, DÏAlessio, & Lizano 1995)
Substituting, one Ðnds
4 CT r2 Lw
Lw
0
] 2r2w [ 2r2w [ bbrb`2
\0 .
7 pT 4 Lt
Lf
0
Then
4 CT
0 \ bbrb .
(21)
7 pT 4
0
The speciÐc heat is proportional to the surface mass
density, which we assume does not change while the perturbation evolves. For an unperturbed temperature given by
equation (5) and using equation (8), the values of the constants are b \ 3/14 and b \ (8 )C T /pT 4, and f is given by
3 1 1
1
8 C
f\
,
(22)
3 pT 3
0
which is proportional to the cooling timescale, given by the
energy content of the disk (per unit surface) CT , divided by
0
the rate at which this energy is lost (per unit surface),
pT 4
0
(the thermal timescale ; Pringle 1981).
Equation (19) then becomes
Lw Lw
.
\
Lf
Lt
(23)
Every function w \ w(f ] t) is solution of this equation ;
then v \ r2w(f ] t). The velocity of the perturbation is the
velocity corresponding to a constant value of the argument
(f ] t), i.e.,
K
R
dR
\[ .
(24)
bf
dt
f`t/const
The negative sign indicates that the perturbation propagates inward, i.e., toward the central star. This happens
because the perturbation in temperature locally changes the
slope of the disk surface, which increases the amount of
irradiation Ñux intercepted by the disk between the position
of the perturbation and the adjacent inner annulus. At the
same time, the slope between the perturbation position and
the adjacent outer annulus becomes smaller than the unperturbed disk slope, which decreases the irradiation Ñux there.
Thus, the perturbation will move toward smaller radii.
The perturbation is damped on a timescale
CA B
v
t \[
\ [v
d
(dv/dt)
D
Lv
Lv ~1
]v
Lt
Lr
fb 2 C(r)T
0,
\
(25)
2
7 pT 4
0
calculated following the perturbation moving through the
disk, which is proportional to the cooling timescale. The
damping time also can be written as t \ R/2v ; the perturbation disappears in roughly half the dcharacteristic time it
would take to reach the star.
Now we can check the validity of our assumptions for the
particular case of accretion disks around low-mass young
stars. The disk temperature can be written as (e.g., Miyake
\
AB
j 2@7
T (r) \ T
r~3@7 ,
0
* 7
(19)
4 CT Lw
Lw
0
\ bbrb
,
(20)
7 pT 4 Lt
Lf
0
and since w is arbitrary, we choose the constants b and b
such that
v\
Vol. 511
(26)
where j, for typical T Tauri stars parameters (e.g., Gullbring
et al. 1998), is given by
A
B
A BA B
A B A B A
kT R 1@2
* *
mGM
*
g
m ~1@2
\ 0.056
3 2m
H
T
1@2 R
1@2 M
~1@2
*
*
*
, (27)
]
4000 K
2R
0.5 M
_
_
where the quantity g is the ratio between the photospheric
height and the disk scale height and m is the mean molecular mass. The photospheric height (in units of stellar radius)
is h \ j(T /T )1@2r3@2.
Assuming*a typical mass accretion rate and central star
properties for T Tauri stars and a typical viscosity parameter a, the damping time can be written as
j\g
B
A BA B A B
A B A B AB
A B
M
15@14 R
~29@14
*
*
0.5 M
2R
_
_
a ~1
T
~32@7 g ~8@7
*
]
0.01
4000 K
3
M0
t \ 0.16r3@14
d
10~8
m 4@7
yr ,
(28)
2m
H
where the mass accretion rate is in M yr~1. The viscous
_
timescale can be written as (Pringle 1981)
]
R2)
t \ a~1
vis
c2
s
A B A B
A B A B AB
A B
M
9@14 R
5@14
*
*
0.5 M
2R
_
_
a ~1
T
~8@7 g ~2@7
*
]
0.01
4000 K
3
\ 233r13@14
m 8@7
yr .
(29)
2m
H
This is comfortably longer than the damping timescale for
any reasonable parameters, so our assumption of constant
C(r) (constant &) is justiÐed, since & can only change on the
viscous timescale (Pringle 1981).
On the other hand, the assumption that the vertical structure responds quickly to the change in irradiation is not
justiÐed at all points in the disk. The vertical disk structure
can respond to changes in temperature on a timescale H/c ,
where c is the sound speed in the disk. Since H/R \ c /v s ,
s orb
where vs is the local Keplerian velocity, the hydrostatic
orb
equilibrium timescale for response is t \ )~1 (Pringle
h
1981). Numerically,
]
A B A
B
R
3@2 M
~1@2
1
*
*
yr .
t \ \ 2 ] 10~4r3@2
h )
2R
0.5 M
_
_
(30)
No. 2, 1999
IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS
FIG. 1.ÈComparison between the damping time of a small perturbation in temperature calculated with the general treatment described in
° 3 (solid line) and calculated with an approximated power-law distribution
for the temperature of a Ñared disk, valid for distances larger than the
stellar radius, discussed in ° 2 (dot-dashed line), with di†erent timescales of
the disk : the viscous timescale (dotted line) and the hydrostatic timescale
(dashed line). The disk has M0 \ 10~8 M yr~1, a \ 0.01, and the central
star has M \ 0.5 M , R \ 2 R , and T_ \ 4000 K.
*
_ *
_
*
This timescale increases with r faster than the damping
time. This means that there is an outer region where the
damping time of the perturbation is shorter than the hydrostatic equilibrium timescale, and the disk has not enough
time to change its height because of the change in temperature (given by the perturbation). The radius beyond
which this happens is
A B A
A B A
A B
B A B
B AB
M0
7@9 M
11@9 R
~16@9
*
*
R B 1.7
h
10~8
0.5 M
2R
_
_
a ~7@9
T
~32@9 g ~8@9
*
]
0.01
4000 K
3
m 4@9
AU .
(31)
2m
H
For R Z R , there is no coupling between diskh
temperature perturbation and height, and our treatment is
not valid. Nevertheless, one expects that, since the disk
height is not a†ected by the temperature perturbation, the
irradiation Ñux remains constant and the only term that
changes with time in the right-hand side of equation (7) is
the disk radiative cooling (increased relative to the steady
state cooling because of the perturbation itself). As long as
the disk cooling is larger than the irradiation heating, its
temperature will decrease with time until it reaches the
steady state solution. (In this case, there is no reason for the
perturbation to move toward the star.) From equation (7)
with a constant irradiation Ñux, this damping happens in
the cooling timescale, given by t B C(r)T /pT 4, which is
0 pertursimilar to the damping timescaled of the 0moving
bation given by equation (50). Therefore, a perturbation in
temperature in R [ R is damped locally before the disk
h
]
899
can readjust its height to be in hydrostatic equilibrium with
its new temperature distribution.
From equation (31) we can see that the larger the value of
M0 /a, the bigger the radius where t B t and the larger the
d
h
region of the disk where temperature and height are
coupled. But for steady disks, larger M0 implies larger
viscous dissipation Ñux and disk optical depth, and the
assumption of vertical isothermality breaks down, at least
for the inner and optically thick regions of the disk. In this
case, equation (7) has to be modiÐed to account for the
viscous Ñux and also for the fact that the midplane temperature (which is the relevant temperature determining the
gas scale height) is di†erent from the e†ective temperature
(which gives the radiative cooling of the disk). When viscous
dissipation is the main heating source of the disk, a perturbation in height has a small e†ect on the disk heating and
one expects that a perturbation in temperature dies out
locally (as long as the disk mass accretion rate corresponds
to a thermally stable branch of the S curve ; Hartmann &
Kenyon 1996, and references therein). In this case, one
expects that the perturbation will damp in a thermal timescale, given by t B C(r)T /F , where F is the viscous Ñux
vis
vis
(Pringle 1981). d
Figure 1 shows a comparison of the relevant timescales
for a disk with M0 \ 10~8 M yr~1 and a \ 0.01. Note that
t \ t only in the inner disk._However, in these regions the
d has
h an almost Ñat surface (Kenyon & Hartmann 1987),
disk
and there it is not valid to assume equation (2). In this case
we must modify our treatment, as discussed in the following
section.
3.
GENERAL TREATMENT FOR SMALL PERTURBATIONS
As in the previous section, the disk is assumed vertically
isothermal and optically thick to the stellar irradiation. The
mean cosine between the stellar incident radiation and the
normal to the disk surface has a more general form than
equation (2), to account for those regions of the disk with a
photospheric height small compared with the stellar radius.
Then, k is given by (Ruden & Pollack 1991)
k\
1
2
CA
B
D
dh h
4
[ ]
,
3nr
dr r
(32)
where as in the previous section we assume k > 1.
The temperature in the unperturbed steady state disk,
given by equation (4), can be found from
A BC
pT 4
*
2r2
A
BD
4
dh
h
0[ 0
]
3nr
dr
r
\ pT 4 ,
0
(33)
where h is given by equation (3), evaluated at the unper0
turbed temperature
T .
0
We perturb the disk
temperature as in equation (6),
assuming that this perturbation is reÑected instantaneously
in the photospheric height and h B h (1 ] v/2). Neglecting
nonlinear terms in v, the right-hand0side of equation (7),
using equation (1), is
F [ pT 4 B
irr
A BC A B
A B D
pT 4
*
2r2
]
v
4
] 1]
2
3nr
h dv
dh
h
0[ 0 ] 0
[ pT 4(1 ] 4v) ,
0
2 dr
dr
r
(34)
900
DÏALESSIO ET AL.
which can be reduced using the steady state solution
C A
B
D
pT 4
Lv
pT 4 dh
h
*h
*
0 [ 0 [ 4pT 4 v ]
.
F [ pT 4 B
0
irr
4r2 0 Lr
4r2 dr
r
Vol. 511
Since the amplitude of the perturbation is small, we
expect that the solution for v is independent of v itself, so
that
(35)
The equation for the evolution of the temperature perturbation can be written as
Lv
Lv
,
D(r) \ A(r)v ] B(r)
Lt
Lr
(36)
df D(r)
\
,
dr B(r)
(45)
A(r)
d ln a
.
\[
dr
B(r)
(46)
Then
f(r) \
with
D(r) \
A(r) \
A
A
C(r)q
0,
pT 3
*
(37)
B
h
1 dh
0 [ 0 [ 4q4
0
r
4r2 dr
B
h
d ln q
0 [ 4q4 ,
\ 0 1]
0
8r3
d ln r
(38)
h
B(r) \ 0 ,
4r2
P
D(r)
dr ,
B(r)
CP
(47)
D
a(r) \ exp [ A(r)/B(r)dr .
(48)
If / A(r)/B(r)dr decreases with r, the amplitude of the
perturbation a(r) increases with r (i.e., the amplitude
decreases when the perturbation moves toward the star).
Since f(r) is a positive quantity the perturbation moves
inward and its (phase) velocity is given by the velocity
dR/dt, which corresponds to the argument z \ (f ] t) \
constant, i.e.,
(39)
v\
R
* .
(df/dr)
(49)
T
q \ 0.
(40)
0 T
*
From the result discussed in ° 2, we look for a solution of
the form
As we have mentioned before, C(r)q P r3@2. Then v P
0 distance to the
q1@2r, and the velocity decreases when the
0
central star decreases. The damping time is given by
v(r, t) \ a(r)w(z) ,
R B(r)
D(r)
df d ln a
\[ *
\[
.
t \
d dr dr
vA(r)
A(r)
(41)
where a(r) is the amplitude of the perturbation as a function
of distance to the central star, which does not depend
explicitly on time. The function w(z) is related to the initial
shape of the perturbation, and its argument z depends
explicitly on time and position, z \ t ] f(r). The quantity
f(r) is related to the velocity of propagation of the perturbation : the radius where the function w(z) is maximum
changes in time maintaining z constant. In those regions of
the disk where the Ñaring is important, we expect that
a(r) B r2, as we have discussed in ° 2.
Then
Lv
dw Lz
dw
\ a(r)
\ a(r)
,
Lt
dz Lt
dz
dw df
da
.
w(z) ] a(r)
dz dr
dr
dw
\ A(r)a(r)w(z)
dz
] B(r)
C
From equation (33), the unperturbed temperature can be
approximated by two di†erent power laws, one for the
outer and Ñared regions of the disk and one for the inner
regions, where the photospheric height is smaller than the
stellar radius. For the outer disk, T P r~3@7, q4 B h /7r3,
0 of
and equation (48) gives a(r) \ r2, and0 the phase 0velocity
the perturbation is v B 7RpT 4/4C(r)T , as before (° 2). On
0
the other hand, for the inner0Ñat regions
of the disk, the
unperturbed temperature is given by q B (2/3n)1@4r~3@4
and the photospheric height is h B h 0r9@8, where h \
2
h (2T /n)1@8. Then the amplitude of0 the 2perturbation trav1
*
eling through those regions is
a B exp [[Er~9@8] ,
(43)
where E B 9.5/nh , and we have assumed that h [ 1.
0
Figure 1 shows2 the damping time for the perturbation,
calculated using equation (50) for the unperturbed temperature obtained by numerically integrating equation (33).
For the plotted models, the damping time calculated
assuming a completely Ñared disk (see ° 2) is a good approximation for R Z 1 AU, but for R [ 1AU, t decreases more
d equations (2)
rapidly as distance decreases. One expects that
and (32) give the same result for radii where
Therefore, the equation for the evolution of the perturbation is
D(r)a(r)
A
D
da
df dw
.
w(z) ] a(r)
dr
dr dz
(50)
(42)
Lv da
dw Lz
\
w(z) ] a(r)
Lr dr
dz Lr
\
N
(44)
B
dh h
4
[ ?
.
dr r
3nr
(51)
(52)
We estimate the radius r beyond which the disk can be
f
considered Ñared for the calculation
of the irradiation Ñux,
No. 2, 1999
IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS
taking the Ñat disk irradiation Ñux equal to a factor f times
the Ñared disk irradiation Ñux. This gives
AB A B
j ~8@9 2 7@9
,
r B f ~7@9
f
7
3n
(a)
0.4
(53)
0.2
and for the typical parameters we have been using and
taking f \ 0.5, r B 40. For 0.5 \ f \ 2, 12 \ r \ 40, so
f
f
this can be considered the transition region between the Ñat
and Ñared disk description. For smaller distances, r \ 12,
the slope of the scale height is small and the disk is Ñat.
When this happens, the perturbation in the disk height has
a negligible e†ect on the irradiation Ñux (which can be
approximated by the irradiation Ñux of a Ñat disk, i.e.,
F B 2pT 4/3nr3). The disk height and temperature are
irr
*because a change in height does not a†ect the
decoupled
heating of the disk. When the perturbation reaches this
region, its amplitude decreases with distance exponentially
(as is shown by eq. [51]), and it cannot move radially any
longer.
0
NUMERICAL EVOLUTION
4.
C(r)
A
B
LT
LT
[ pT 4 ,
\ F r, T ,
irr
Lr
Lt
(54)
where we have written explicitly the spatial derivative of T
in the argument of the irradiation Ñux. We assume that the
disk is always in hydrostatic equilibrium. The shadowing is
introduced by taking F (r) \ 0 for r [ r , if h(r)/r \ h(r )/r .
irr has a local maximum,
t
t
The radius r is where h(r)/r
and tit is
t
given by the radius where the slope (L ln h/L ln r) \ 1 and
rt of the
L2h/L2r \ 0. At r \ r a straight line from the center
t
star is tangent to the disk surface.
We calculate numerically the unperturbed steady state
temperature distribution integrating equation (33). At a
time t \ 0, this disk is perturbed by
C
D C
D
a(r )r2
(r [ r )2
0
0
exp [
,
(55)
2p2
r2
0
v
where r is the perturbation initial position, a(r) \ r2a(r )/r2
0
0 0
is its amplitude,
and p is its initial width. We use the Macv
Cormack predictor-corrector method (MacCormack &
Paullay 1974) to integrate equation (54). The time step used
in the integration is dt \ 1.25 ] 10~5 yr, and the spatial
grid consists of 300 points, logarithmically distributed,
between r \ 1.36 and 2000. The disk has M0 \ 10~8 M
yr~1, a \ 0.01, and the central star has M \ 0.5 M _,
*
_
R \ 2 R , and T \ 4000 K.
*Figure _
*
2 shows the evolution of a perturbation initiated
at r \ 200, with p \ 0.1r , and a(r ) \ 0.15 and 0.5. The
0 position of the
v perturbation
0
initial
is 0close to the outermost
radius at which the hydrostatic timescale is short enough to
assume the disk is in hydrostatic equilibrium. The elapsed
time is t \ 1 yr, and each curve corresponds to a time step
*t \ 0.1 yr. The Ðgure shows the decreasing amplitude of
the perturbation while it moves toward the star. The position of the maximum of v is independent of its initial amplitude, as can be seen comparing Figures 2a and 2b. For
comparison, Figure 2c shows the evolution of the highv\
(b)
0.4
0.2
0
(c)
0.4
0.2
0
0
In this section, we calculate numerically the evolution of
an arbitrary size temperature perturbation, including the
e†ect of the shadowing of the disk caused by the perturbation.
The equation we integrate is equation (7), i.e.,
901
100
200
300
400
r
FIG. 2.È(a) Perturbation as a function of position for di†erent times
(*t \ 0.1 yr). Initially, the perturbation is a modiÐed Gaussian, centered in
r \ 200, with a width equal to p \ 0.1r (dotted line). The disk has a mass
0
p a viscosity
0
accretion
rate M0 \ 10~8 M yr~1,
parameter a \ 0.01, and it
_
surrounds a star with M \ 0.5 M , R \ 2 R , and T \ 4000 K. The
*
_is v(r
* ) \ 0.5.
_ (b) The* evolution of the
initial amplitude of the perturbation
0 (a), but an initial amplitude
perturbation for the same parameters as
v(r ) \ 0.15. (c) The evolution of the perturbation with amplitude v(r ) \
0
0.5,0 calculated assuming the irradiation Ñux is constant in time and equal
to the irradiation Ñux of the unperturbed disk (see text).
amplitude perturbation if the irradiation Ñux remains constant. This is an example of what happens in those regions
where the hydrostatic timescale is larger than the temperature perturbation damping time. We do not include the
e†ect of the shadow in this last case, since the perturbation
in temperature has no e†ect on the shape of the disk. Comparing this result with the evolution of a perturbation with
the same initial amplitude but a changing surface shape, we
see that the moving perturbation decreases faster than the
static one. For instance, after t \ 1 yr the moving perturbation has a value of the maximum of v smaller by a factor
D0.6 with respect to the static one because when the perturbation moves toward the star, the radiative cooling rate
increases more than the irradiation heating rate.
Figure 2 shows that behind the perturbation there is a
region with v \ 0, i.e., with a temperature lower than the
unperturbed steady state temperature, which corresponds
to the shadow. The external boundary of the shadow moves
toward smaller radii, while the perturbation evolves. This is
shown by Figure 3, where h/r is plotted as a function of
distance for di†erent time steps. From Figure 2 we see that
the e†ect of the shadow, as expected, is larger for the larger
initial amplitude. The minimum value of v (related to the
shadow) is D[0.15, around r \ 250, for the perturbation
with a(r ) \ 0.5 initiated at r \ 200. The temperature at the
0 region decreases with time, with a thermal timeshadowed
scale proportional to the local damping time. As we have
shown in ° 2 and ° 3, this timescale increases with distance
(see Fig. 1). Therefore, the bigger the shadow, the longer the
time its outer regions need to decrease their temperature
because of the decreasing heating. But the timescale in
902
DÏALESSIO ET AL.
FIG. 3.ÈRatio between the photospheric height and the radial coordinate h(r)/r as a function of r for a perturbation with a(r ) \ 0.5, at r \ 500.
0
0 t\2
The time t \ 0 is plotted with a dotted line. The last plotted
time was
yr and *t \ 5 ] 10~2 yr. Disk parameters are the same as in Fig. 2. The
shadow is produced at those outer regions where h/r is smaller than the
maximum value of h/r at the bump.
which the perturbation evolves is smaller than the timescale
in which the shadows evolve (because the perturbation is
always located at a smaller radius than the shadow it
produces). Figure 3 shows that the perturbation is damped
fast enough to prevent a large e†ect of the shadow on the
disk structure.
We calculate the radius r
where the function v/r2 is
maximum. According to ° 2, max
for a small perturbation in the
FIG. 4.È(a) Amplitude (v/r2) as a function of the radius r . Each curve
maxwith a di†erent initial amplitude
max and posicorresponds to a perturbation
tion : a(r ) \ 0.5, r \ 200 (solid line), a(r ) \ 0.5, r \ 500 (dotted line),
0
0
a(r ) \ 0.15,
r \ 1000
(dashed line), a(r )0\ 0.15, r0 \ 5000 (dot-dashed
0
0
0 of the maximum of
line). In each case, we take p \ 0.1r . (b) 0The position
0 perturbations plotted in (a).
v/r2 as a function of time, for vthe same
Vol. 511
Ñared region of the disk, w(f ] t) \ v/r2. Therefore, r
is
max
the position of the maximum of w(f ] t) and w(r ) \ const
max
in those regions where the assumptions used in ° 2 are valid.
Figure 4 shows w(r ) \ (v/r2)
as a function of r ,
max
rmax
for perturbations withmaxinitial amplitudes
of 0.15 and 0.5,
initiated at r \ 200, 500, and 1000. In each case there is a
0
range of r
for which the amplitude of the perturbation is
max
approximately proportional to r2 as we expect from equation (16) (see ° 2). For the large amplitude perturbations, at
larger distances (or shorter times) the perturbation is not
small enough to be described by the linear equations discussed in °° 2 and 3, and its amplitude decreases with position (or time) faster than the linear prediction. There is an
inner region around r D 100 where the irradiation
max
changes because the disk is almost Ñat and the damping
time decreases faster than the linear prediction (see Fig. 1).
When the distance between the perturbation and the central
star is similar to the width of the perturbation, the presence
of the inner boundary is reÑected in the sudden decrease of
w(r ) toward smaller radii.
max
5.
CONCLUSIONS
We have considered the evolution of a temperature perturbation in a vertically isothermal disk in vertical hydrostatic equilibrium, irradiated by the central star. We Ðnd
that there are three main regions whose details depend on
the speciÐc parameters of the disk and the star :
1. The outer disk, for R Z 2 AU for typical parameters,
where Ñaring is important in determining the disk unperturbed temperature, but where the hydrostatic timescale is
longer than the perturbation damping time. In this region
the disk temperature and scale height are decoupled : the
disk has not enough time to reach the hydrostatic equilibrium corresponding to the new temperature before radiative cooling damps out the perturbation. The perturbation
is damped in a cooling timescale (D3.7t ), between D0.7
d and a typical
and 10 yr, for R between 2 and 1000 AU
T Tauri accretion disk.
2. The intermediate disk, for 0.1 Z R Z 2 AU, where the
temperature distribution is changing from the ““ Ñat ÏÏ to the
““ Ñared ÏÏ characteristic temperature distributions, but where
the damping time of the perturbation is longer than the
hydrostatic timescale. In this region the disk is always in
hydrostatic equilibrium and the temperature and scale
height are coupled. The perturbation in temperature (and
photospheric height) moves toward the star, with decreasing amplitude, in timescales between 0.001 and 0.1 yr, for
typical parameters.
3. The inner disk, for R [ 0.1 AU, where the disk photospheric height is smaller or similar to the stellar radius and
the temperature distribution is similar to that of a Ñat irradiated disk. The perturbation moves toward the central star
and its amplitude decreases exponentially.
For a disk in hydrostatic equilibrium, a shadow caused
by a bump at the disk surface produces a variation in the
disk-temperature structure that evolves more slowly than
the bump itself. The consequence of this is that the original
perturbation disappears before the shadow can produce a
large e†ect on the disk structure and after a time similar to
the perturbation damping time the disk returns to its original, unperturbed structure.
No. 2, 1999
IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS
We have neglected some e†ects that could be present in a
more realistic disk model. For instance, as discussed in ° 1,
disks where irradiation dominates may have temperature
inversions. Nevertheless, we do not see why the modest
change in the local scale height should produce qualitatively di†erent results. Similarly, we have ignored radial
transport of energy and accretion viscous heating. These
two e†ects should help stabilize the disk further because
they tend to smooth the temperature distribution.
Since the rippling of the disk arising from temperature
inhomogeneities dies out in a time shorter than the viscous
timescale, we conclude that the irradiated disk is stable
(Cunningham 1976) and, as in Blair et al. (1984), that a
903
smooth surface can be assumed to calculate the Ñux and
incidence direction of the stellar irradiation in our detailed
modeling of the disk vertical structure (e.g., DÏAlessio et al.
1998).
We are grateful to John Raymond for making us aware of
the Cunningham (1976) paper and to Alejandro Raga for
fruitful discussions. This work was supported in part by
Instituto de Astronom• a, UNAM, Mexico, DGAPAUNAM, ConaCyT, and NASA grant NAG5-4282. P. D.
thanks Harvard-Smithsonian Center for Astrophysics for
their hospitality, and was supported in part by the Visitor
Scientist Program of the Smithsonian Institution.
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