Download L7 Protoplanetary disks Part III

Lecture 7
Protoplanetary disks
Part III
Lecture Universität Heidelberg WS 11/12
Dr. C. Mordasini
Based partially on script of Prof. W. Benz
Mentor Prof. T. Henning
Lecture 7 overview
1. Viscous accretion disks continued
1.1 Analytical solution: steady state
1.2 Analytical solution: time dependent
1.3 Temperature structure
1.4 Energy budget and SEDs
2. Disk dispersion by photoevaporation
2.1 Internal photoevaporation
2.2 External photoevaporation
3. Central magnetospheric cavity
1.1 Analytical solution:
steady state
Steady state
We recall the master equations for the angular momentum transport:
T=torque
j=specific ang. momentum
In general, there is no analytical solution to the equation as it is non-linear. Lyden-Bell & Pringle
1974 have worked out a number of analytical solutions to the master equation under simplifying
assumptions. We study three of them with increasing complexity.
Steady state solution with zero torque inner boundary condition
A special solution is the simple case that the mass flux is constant as a function of radius, i.e.
In that case, the master equation can simply be integrated on both sides:
To determine the integration constant T0, we assume that the star is rotating slower than with
breakup speed. This means for the angular frequency of the star’s rotation at the surface
Steady state II
Then, there must exist a boundary layer at Rc where the disk's Ω changes from the (faster)
Keplerian value to the slower stellar value. i.e. where
Ω
T=0
The torque is proportional to the gradient of the angular
velocity, therefore the torque vanishes at the place where
Ω turns over at Rc. Therefore,
Ω★
R★
at
Rc
R
where the second approximation holds for a small viscosity, i.e. a negligible thickness of the
boundary layer.
The specific angular momentum is simply
It should be noted that
is the angular velocity of the star grazing orbit, and not the real
angular velocity of the stellar surface (which is as said smaller than this).
We therefore find for the steady state solution
Steady state III
With that we can calculate all quantities torque T, mass flux, surface density and radial velocity
of the gas as
Note
Given a viscosity, this equation defines the steady-state surface density profile for a disk with a
constant accretion rate. Away from the boundaries, Σ(r) ∝ ν−1.
Assuming an alpha viscosity, we can make this more explicit:
The link with the scale height means that here, the vertical
structure of the disk comes in, or in other words, the
temperature structure, which is a priori unknown.
Steady state IV
The last equations mean that
.
Suppose we know that Tmid is a given power-law:
Shakura & Sunyaev model
Then, away from the inner boundary,
Examples:
For the passively irradiated disk, we have
Tmid~ r"1/ 2
!
" ~ r#1
!
Dullemond
Note however that this is not self-consistent because we have to take into account viscous
heating, but it illustrates how disk surface density and temperature are linked.
Shakura & Sunyaev model
Examples:
Steady state V
To get a surface density profile like for the MMSN, we must have a constant disk temperature
(unlikely!)
Tmid~ r"0
" ~ r#3 / 2
!
!
Dullemond
In the last figures, the effect of the inner boundary was neglected. Taking this into account, we
find
The radius of the sun is today
about 0.0046 AU. Initially, it
was larger by a factor of a few.
1.2 Analytical solution:
time dependent
Time dependent solutions
Time dependent solution for a point mass without central couple
We now look at time dependent solutions in a fixed outer gravitational field. The change of
the gravitational field due to accretion onto the central object is thus neglected.
Rewriting the master equation, using the continuity equation in cylindrical coordinates and
assuming that ν(r) varies as a powerlaw of r, yields the following partial differential equation:
where for the central point mass:
Solutions T(j,t) or equivalently T(r,t) to this differential equation can be resolved in modes in
which the variation of T with the time t is proportional to
General solutions are then found as a superposition of such solutions, where their specific form
is given by the boundary conditions.
For the case of T(h=0,t=0)=0 (no central couple), and the initial condition
Time dependent solutions II
It is found after some algebra that it is convenient to use in this case the dimensionless time
tν =1 corresponds thus to the initial time. The solution is then given as
Radius of maximum viscous couple (or radius of velocity reversal)
From the flux or vr it can be seen that for
inwards inside the corresponding radius
the velocity is outwards, and
This position is also the position of maximum viscous couple (torque) as we can easily determine
from finding the maximum of T(r,tν).
Time dependent solutions III
Time dependent solution for a point mass with central couple
We can use the result from the steady state case to construct a time dependent solution with
central couple. The equations in the last section indicate that near the center, the time dependent
solution has the same form as the steady state solution without a central couple:
Thus, we assume that if we now want to include the central couple with the star, we can
replace this expression with the one for the steady state with central couple,
leaving the rest unchanged.
For a physically usable solution, we must determine the constants a and C.
The constant a comes from the initial profile of the disk, and is related to the initial size of the
disk R1 (the place where the surface density is e-0.5 the value at the center in the solution
without central couple) as
The constant C is found from the total disk mass:
We find:
Time dependent solutions IV
Putting these expression together we finally find:
(r>>R )
★
The dimensionless time can now be written as
where
is the viscous timescale measured at R1. We also have:
Example
Parameters
Disk mass [Msun]
0.020
0.018
0.016
0.014
0.012
0.010
0.008
0
2
4
6
8
10
Stellar accretion rate [Msun/yr]
t [τ]
Notes
•Disk mass and accretion rate gradually decrease.
•The process is fastest at the beginning.
•The numerical values are similar to observed ones
(for the chosen value of e.g. the viscosity).
1 � 10�7
5 � 10�8
2 � 10�8
1 � 10�8
5 � 10�9
2 � 10�9
1 � 10�9
0
2
4
6
t [τ]
8
10
Surface density evolution
t=0, 0.1, 0.3 and 1 τ
200
Σ [g/cm2]
Σ [g/cm2]
100
50
20
10
a[
AU
]
τ]
[
t
0
10
20
30
a[AU]
Note
-disk is getting larger in time.
-surface density is decreasing in time.
40
50
Gas velocity evolution
t=0, 0.1, 0.3 and 1 τ
t [τ]
400
vr [cm/s]
vr [cm/s]
200
0
�200
�400
1
5
10
50
100
500
a[AU]
a[AU]
50
RMVC [AU]
Note
•inner part of disk is accreting.
•outer part is decreting.
•the point of velocity reversal at RMVC moves itself
outwards as the time increases, overtaking more and
more material (it moves outwards faster than the gas
itself). Gas initially not much outside RMVC first flows
outwards, and inwards somewhat later on.
•in the end, an infinitesimal amount of mass is at infinity,
containing the angular momentum.
40
30
20
10
0
2
4
6
t [τ]
8
10
Gas flux [Msun/yr]
Gas flux evolution
t [τ]
a[AU]
1.5 � 10�7
We see a wave like feature
traveling outwards,
decreasing in amplitude.
Gas flux [Msun/yr]
1. � 10�7
5. � 10�8
0
�5. � 10�8
�1. � 10�7
�1.5 � 10�7
1
2
5
10
20
a[AU]
50
100
200
1.3 Temperature structure
Temperature
To get the temperature of the disk, we consider the heating by viscous dissipation and the
cooling at the disk surface by radiation.
Let us compute the vertically integrated rate of viscous dissipation [erg/ (s cm2)] which is
given as the vertical integral over stress times rate of strain
T = 2πr
�
∞
rσrϕ dz
−∞
For the steady state disk, we have far from the inner boundary for the value of the torque
so we find (quasi Keplerian)
This heat must be transported away. Since for a thin disk the z gradient dominates the radial
gradient, the heat equation is well approximated by
as the vertical heat transport inside the disk can happen through several mechanism.
Temperature II
At the two disk surfaces, we must finally have cooling as a black body:
For the thin steady accretion disk far from the inner boundary this gives
Midplane temperature
Teff is the temperature at the surface. If the heat transport is radiative, and the disk is optically
thick, we can estimate the midplane temperature which depends on the optical depth.
The optical depth is defined as (κRoss is the Rosseland mean opacity)
The basic equation for vertical radiative flux F(z) in the diffusion approximation is:
For simplicity, we assume that all energy is liberated at z=0 (in the midplane). Then, F(z)
is constant with z and given as
. Assuming that κRoss is also constant with z,
we can integrate the differential equation.
Temperature III
For the second line we have used that for τ ≫ 1 almost all of the disk gas lies below the
photosphere. For large τ, Tmid4 ≫ Teff4, and the last line simplifies to
We finally find for the midplane temperature due to viscous dissipation
This equations provides the requested link of temperature (vert. structure) and surface
density (radial structure).
For full solutions, we first need an expression for the Rosseland mean opacity.
We can then express the midplane temperature as a function of the surface density.
In the radial structure equations we can then eliminate the temperature (e.g. in the viscosity)
in exchange for the surface density.
We then have an equation entirely in terms of the surface density which we can solve.
Temperature IV
In general, such solutions can only be calculated numerically, as e.g. the opacity is not a
simple analytical function.
- α-disk
- viscous heating only (Alibert et al 2004)
- opacity transitions lead to different regimes
- temperature set by accretion rate
- this also sets the position of the ice-line,
which is important for giant planet formation.
- α-disk
- includes both viscous dissipation and stellar irradiation
(Fouchet et al 2011)
- and photoevaporation (Mordasini et al. 2011)
- alpha 7 x 10-3
- temporal evolution of the central temperature
(one line all 20‘000 years).
- as the disk mass decreases, T approaches the
irradiation only profile.
1.4 Energy budget and SEDs
Energy budget
We can compute the amount of energy dissipated in the disk between the inner and the outer
boundary:
where to obtain the last equality we assumed a Keplerian disk.
Note that this energy is equal to half the total gravitational energy, the outer half has gone
into orbital kinetic energy. This kinetic energy must be dissipated in the boundary layer near
the star. In other words, the equivalent of the entire energy dissipated in the disk must be
dissipated in the boundary layer.
Since this region is narrow, the energy dissipated heats up the layer to high temperatures so
that short wavelength radiation is generally associated with the boundary layer (UV excess).
"F"
Wien
region
Spectral energy distribution SED
SEDs provide important
RayleighJeans region
observational
information
on the structure of protoplanetary disks.
Each annulus of the disk radiates as a black body. The maximum of the radiation of a black
body at temperature T is found according
! to Wiens law at
Dullemond
The observed SED is the
superposition of the
contributions of all annuli in the
multi-color region.
At short wavelength, there is the
Wien region, while at long
wavelength, there is the
Rayleigh-Jeans regions,
SED II
A disk with a power law temperature distribution that extends over a large enough distance
tends to radiate a power law spectral energy distribution. To see this, we realize that each
optically thick annulus 2 π r dr contributes mostly with photons from the peak of the local black
body distribution. From Wien’s law, we have for the steady state disk
Thus, the emergent radiation will have the form
and therefore
For the general case, one can show that for
we have
Multi-color blackbody disk SED
Rayleigh-Jeans region:
!F!
!(4q-2)/q
Slope is as Planck function:
"F" # " 3
Multi-color region:
!3
"
SED III
SED
of
SED of
accr
%
%
3
Remember:
Remember:
While this would provide a well observable signature of a viscous accretion disk, itTis = ' Teff M
=˙ '$
eff
& 8"# flat& 8
unfortunately not the only model that produces such a SED. It was pointed out that a simple
disk reprocessing the light received from a central ball will haveAccording
the same
SED.
This
because
According
toisour
derived
to
our
derived
SED ru
2
in such a case the light drops with the usual 1/r compounded with the geometrical factor
which is proportional to R/r where R is the radius of the central object.
" F" # "
!
!
Does
this fitofSEDs
of Her
Does
this
fit
SEDs
Herbig
Ae/
Observed SEDs from T-Tauri stars
˙ = 2 "10 M /yr
need M
˙
need M = 2 "10 M /yr
!
HD104
!
#7
sun
#7
sun
!
HD104237
!
Adams et al. 1990
Dullemond et al.
While some SED have a shape compatible with the purely viscous accretion disk model,
others have a flatter SED. Flat SED imply a temperature dependance with radius: Teff ∝ r-1/2
as opposed to the steeper dependance derived for the purely viscous disks with Teff ∝ r-3/4.
Distinguishing passive irradiation from active viscous accretion is difficult from the SED.
2. Photoevaporation
Photoevaporation
Self-similar solution of a viscous disk: surface density and accretion rate decline as powerlaws at late times: => Smooth transition between disk and diskless states expected.
Not observed: YSO between CTTS and WTTS (so called “transition objects”) are rare. From
observed statistics, one infers that the time scale for clearing the disk is short, only ~105 yrs.
An additional mechanism besides viscosity is needed to allow such a behavior.
An observational indication to the mechanism that additionally could drive disk dispersal was
provided by HST observations of low mass stars exposed to the strong ionizing flux
produced by neighboring massive stars (O-stars) in the core of the Orion Nebula’s Trapezium
cluster.
The images show drop shaped nebulae surrounding young stars with protoplanetary disks,
which are interpreted as the signature of photoevaporation and escape of disk gas.
Observations of photoevaporation
Basic mechanism
•Irradiation of disk surface creates a thin surface layer of hot gas with T>Teff.
•If its temperature/thermal velocity exceeds the local escape velocity, the surface layer gets
unbound and evaporates, i.e. a thermal wind is launched taking away disk gas.
Basic types depending upon origin of the disk-irradiating flux
1) External photoevaporation: the radiation originates from other stars than the host star, usually
nearby massive stars in young clusters that have prodigious ultraviolet luminosities as it is the cae
in Orion.
2) Internal photoevaporation: for disks around stars forming in groups or small clusters (e.g.
Taurus) the influence of external photoevaporation is small (no nearby massive stars), and mass is
instead driven internally, by the radiation from the central star itself.
Relevant radiation types
1) FUV: Far-ultraviolet radiation. 6 eV < hν < 13.6 eV. Dissociates hydrogen molecules but does
not ionize hydrogen atoms.
2) EUV: Extreme-ultraviolet. 13.6 eV < hν < 100 eV. Ionizes hydrogen.
3) X-rays. Defined by convention as photons having hν > 0.1 keV.
Internal photoevaporation
Armitage 2010
For internal photoevaporation,
irradiation by all three types of
radiation can play a role. Here we
focus on EUV driven evaporation
as modeled by Clarke, Gendrin &
Sottomayor 2001.
The host star EUV flux leads to a temperature of the ionized disk surface layer of
The mean molecular weight of the ionized gas μII is about 0.6.
This leads to a speed of sound in the gas
The mean thermal velocity is similar to the sound speed.
.
Internal photoevaporation II
Gas with this temperature is unbound from the star beyond the so called gravitational radius
For 1 Msun star, the gravitational radius is at about 7 AU. Numerical simulations show that some
mass is already lost at somewhat smaller distances beyond
.
The resulting evaporation can then be estimated as
where n0(r) is the number density of ionized atoms at the base of the heated layer.
The latter is found in detailed photo-hydrodynamical simulations as
(1)
(2)
(3)
In equation (2), kH is a constant, and Φ41 is the stellar output of ionizing photons in units of 1041
photons per second. The origin of this radiation comes from both the background photospheric
flux, and the hard radiation from the stellar boundary accretion layer. The latter establishes a link
with the viscous evolution of the disk.
Internal photoevaporation III
Note the square root dependence of the number density on Φ41 in (2). It comes from the fact
that the flow is ionization/recombination limited. This means that most of the photons are
expended maintaing the ionization level in the flow, and only a small fraction is used to create
the mass flux from the interface between the neutral/ionized interface at the disk surface.
The equilibrium between the photoionisation rate of neutral atoms with a number density nbase
and the radiative recombination rate of ionized atoms with density n0 at the surface means
[1/(cm3 s)]
where
is the photoionisation cross section of H for UV radiation [cm2]
while αrec is the radiative recombination coefficient for hydrogen ions [cm3/s]. Therefore,
. Equation (3) finally means that the flow is mostly concentrated near Rwind.
For typical parameters (kH=5.7x107 , Φ41=1, βII=0.69 i.e. Rwind=5 AU for 1Msun; Hollenbach et
al. 1994) this gives a constant mass loss of about 3 x 10-10 Msun/yr.
As we shall see below, this becomes important towards end of disk evolution, and causes the
formation of a hole/gap in the disk at about Rwind.
Internal photoevaporation IV
Alexander et al. 2005
Illustration of a photohydrodynamic simulation,
showing a wind flowing away
from the inner disc due to
internal photoevaporation.
Density is plotted as a color
scale, with the ionization front
denoted by a dashed line and
the computational boundaries
denoted by solid lines.
Velocity vectors are plotted at
regular intervals.
The plot shows a state late in the disk evolution, when the inner disk (inside of the gap) has
already disappeared. The disk is now directly irradiated by the star and cleared in an inside-out
way.
External photoevaporation
Very close to massive stars (<0.03 pc) external EUV photoevaporation is important. At more
typical (larger) distances, FUV radiation from the massive stars dominates, as it is less strongly
attenuated by the background gas in the stellar cluster.
This external FUV causes the formation of a hot neutral layer on the disk’s surface (Matsuyama,
Johnstone & Hartmann 2003). The mass loss rate is now estimated as:
The gravitational radius is now of order Rg,I≈ 140 AU. Mass loss occurs outside this radius.
Here, rmax is the outer disk radius, β=0.5 and the total mass loss rate is a parameter which is
depends e.g. on the distance from the massive star providing the heating radiation.
This process takes away mass outside about 70 AU. It is important from the beginning by
reducing the external disk radius and by mass loss there.
Combined model
Surface density [g/cm2]
10000
’fort.14’
βRg,Iu 2:3
Rwind
Combined viscous evolution and
internal & external photoevaporation.
1000
100
α parameter of 7 x 10-3.
10
Viscous dissipation and stellar
irradiated included for the temperature
structure.
1
0.1
1-2 x MMSN disk (Mdisk(t=0)=0.024 M⊙
0.01
0.001
Mean Ṁwind,ext=7 x10-9 Msun/yr.
0.1
Mordasini et al. 2011
1
10
Semimajor axis [AU]
100
1000
Resulting disk lifetime τ=2.01 Myrs.
A red line shows the gas surface density all 20000 yrs. The arrows show the direction of the
mass flux at the end state.
Combined model II
Characteristic phases
1. The initial exterior radius specified by the initial conditions gets quickly reduced by external
photoevaporation to a radius where mass removal due to photoevaporation, and the viscous
spreading of the disk are in a quasi-equilibrium as described. In the specific case, the radius
decreases from initially about 200 AU to ∼ 100 AU. In the inner part, the disk very quickly
evolves from the initial profile towards equilibrium.
2. In the second, dominant phase a quasi self-similar evolution of the disk occurs. The inner part
of the disk (r ≤ 10 AU) is in near steady state (i.e. the mass accretion rate is nearly constant as a
function of radius), and the slope of gas surface density γ is approximately -0.9 (but varying
between -0.4≤ and -1.5 due to opacity transitions). The outer radius is slowly decreasing, here
from about 100 AU down to 60 AU. For more massive disk, this equilibrium radius is further out.
3. Once the viscous gas accretion rate in the inner disk becomes comparable to the mass loss
rate due to internal photoevaporation (happens when the surface density has fallen to about
0.01 to 0.1 g/cm2 at ∼10 AU), a gap opens somewhat outside of Rwind. The evolution of the
disk now speeds up which correspond to the so called “two-timescale” behavior (Clarke et al.
2001).
4. Quickly afterwards, the total disk mass has fallen to 10−5 M⊙, where the calculation is
stopped. The effect of the direct radiation field which would clear the disk quickly from inside
out once the gap has opened is not included in the simulation shown here.
3. Central magnetospheric cavity
Central part of the disk
The central part of the protoplanetary disk (a<0.1 AU) has a complicated structure as
temperatures are high (dust evaporation, ionization), stellar irradiation is strong and stellar
magnetic fields start to become important. We here focus on the aspect of magnetospheric accretion (cf. Bouvier et al. 2006).
The final accretion of gas onto the star (and ejection via jets) takes place in the compact
magnetized region around the central star. The inner disk edge is interacting with the star’s
magnetosphere, leading simultaneously to magnetically channeled accretion flows and to
high velocity winds and outflows.
The magnetic star-disk interaction is thought to have strong implications for the angular
momentum evolution of the central system, the inner structure of the disk, and possibly for
halting the migration of young planets close to the stellar surface (Hot Jupiters).
Surface fields of order of 1-3 kG have been
derived from Zeeman broadening
measurements of CTTS photospheric lines.
The plot shows measured stellar magnetic field
strength of T Tauri stars in Orion, Taurus/Auriga
and the TW association, as a function of time.
Yang et al. 2011
Basic structure
Assuming that T Tauri magnetospheres are
predominantly dipolar on the large scale, we
show below that the inner accretion disk is
expected to be truncated by the
magnetosphere at a distance of a few stellar
radii above the stellar surface for typical
mass accretion rates of 10−9 to 10−7 M⊙/yr.
Disk material is then channeled from the disk
inner edge out of the plane onto the star
along the magnetic field lines, giving rise to
magnetospheric accretion columns.
The free falling material in the funnel flow eventually hits the stellar surface where an
accretion shock develops near the magnetic poles of the star.
Size of the cavity
The size of the magnetospheric cavity can be estimated by assuming a spherical accretion flow
onto the star with free fall velocity, and a dipolar magnetic field.
The ram pressure of the accreting material is
This will at some point be offset by the magnetic pressure for a sufficiently strong stellar field.
Where these two pressures are equal, if the accreting matter is sufficiently ionized, its motion
will start to be controlled by the stellar field. This point is usually referred to as the truncation
radius RT.
If we consider the case of spherical accretion, we have
For a dipolar stellar magnetic field we also have
Size of the cavity II
Equating the two pressures and solving for R gives
For typical values this corresponds to
where B3 is the stellar field strength in kG,
is the mass accretion rate in units of 10−8
M⊙ /yr, M0.5 is the stellar mass in units of 0.5 M⊙, and R2 is the stellar radius in units of 2 R⊙.
Notes
-the magnetospheric cavity has thus the size of about 7 stellar radii, or roughly 0.05 AU.
-a strong magnetic field makes a larger cavity.
-a low gas accretion rate makes a large cavity. This could indicate that the cavity expands
towards the end of the disk lifetime.
Size of the cavity III
In the case of disk accretion (instead of spherical accretion), the numerical coefficient
above is different, but the scaling with the stellar and accretion parameters remains
identical. In an accretion disk, the radial motion due to accretion is relatively low while the
Keplerian velocity due to the orbital motion is only a factor of 21/2 lower than the free-fall
velocity, i.e. similar.
The low radial velocity of the disk means that the disk densities are much higher than in the
spherical case, so that the disk ram pressure is higher than the ram pressure due to spherical
free-fall accretion. As a result, the truncation radius will move closer to the star. In this regard,
the equation for RT gives an upper limit for the truncation radius.
Therefore, models typically use a inner disk radius of
where k≤1.
In reality, CTTS have more
complex magnetic field
topologies, which need not
to be aligned with the
rotation axis.
In this case, numerical
simulations are necessary to
get the structure of the cavity.
A slice of the funnel stream obtained in 3D simulations for
an inclined dipole (Θ = 15◦). The contour lines show
density levels,The thick lines depict magnetic field lines
(Romanova et al., 2004a).
examine the Lindblad torque for each n because of the
exchange of the angular momentum between vertical modes
with di†erent n. Only the Lindblad torque summed over n is
well deÐned. If the scale height increases with r, this e†ect
-decreases
!
the di†erential torques, as shown in Table 2. AsStrong
a
B
Weak
B
result of an increase in h with r, the !<<0
vertical averaging is
at 0.1 AU
no cavity
more e†ective
for outer THREE-DIMENSIONAL
Lindblad resonances than forcavity
the DISK-PLANET
No. 2, 2002
I
!≈0.9
inner ones. This makes outer Lindblad resonances weak
and decreases the di†erential torques.
is more appr
smaller thanThe
thatthree
in two-dimensional
disks
of the
asymmetric e†ects on
the because
total corotation
the pressure
ine†ectiveness
of !the
gradient.
torque,
(a) pressure
, !(d) , and
!(k) , are listed in Table 3. Since
CT,n torque
CT,n is obtained
CT,n from the solution around
the corotation
To see how
corotation, !(k)5. DISCUSSION
is well deÐned for each n. For n \ 0, as
the two-dim
CT,n
readily seen from equation (55), only !(a) has a nonzero
setting n \ k
CT,0toThe
5.1.
Orbital
Migration
of
Planets
due
the contribuvalue,
and
it
is
given
by
!
(a)
\
[2!
(curv)/3.
CT,0
CT,0
MMSN, irrad., Σ0 ∝ a-0.9, Distance [AU]
equation (21
Distance
[AU]
Disk-Planet
Interaction
tion
from
n
[
0
is
small
in
all
e†ects
as
well
as
in
the
curva,alpha=7x10-3
coordinate x
ture e†ect.
As
stated
in
°
1,
the
radial
migration
of
a
planet
due
to
the
As a zero order approximation,
canresults,
take the
of the cavity
intodi†eraccount by Lindblad
forcing the res
From the we
above
we effect
can calculate
the total
interaction
is distance,
an and
important
process
intorque
planet
surface disk-planet
density to fall
to
zero
at some
as total
shown
on the left.
ential
Lindblad
torque
the
corotation
in
obtain
formation. general
In thethree-dimensional
previous section
obtained combining
the total
disks.we
Furthermore,
wewith
obtain
the
total disk leads to
As we will
see later,
the, Lindblad
gravitational
oftorques,
a planet
the through
gaseous
torque,
! the
exerted and
on interaction
acorotation
three-dimensional
disk
total
netplanet,
torque
on the itdisk,
! , for
three-dimensional
disks
torquesthe
acting
on
the
causing
to Since
migrate
radially.
Fortorque
low mass
planets
in so called
disk-planet
interaction.
the
total
on
a
total
1.1
1.2
two-dimensional
disks as
type I migration,
theand
torque
is:
Σ
Gas surface density [g/cm2]
Gas surface density [g/cm2]
Implication for planetary migration
A
A
B
B
planet has the opposite sign, the radial migration speed of
M r ) 2
where
b4
the planet, r5 , !is given
by
p
p
p
(3D)
\
(1.364
]
0.541a)
p
r
4
)
2
,
p total
p
p p p
M c
]
2)/
/()
c!
p,m
dL ~1
the Lin
Torque(69)
causes change ofnear
planet’s angular
p Tanaka
M rtotal
) ,2
\ [2r
r5 \ L0
rbation excited by the
et al. 2002
, (68)L , i.e. radial migration.
!p (2D)p\dr
(1.160 ] 2.828a) p p Lp p p r4 )2momentum
equation (72
ation is normalized in
p
p
p
total
M
c
p
malized
surface
Thus,
thedensity
inner cavity with α<<0 is ap strong stoppingc mechanism
for low mass planets
would not ch
where
momentum
of
Lthree-is ones in type II.
where
we substituted
k \ 3/2, torque,
d \ the
3/2but
] planet
anot
for for
themassive
migrating
in typethe
I as
itangular
causes
a strong
outward
introduce a
p
dimensional
case
and
d
\
a
for
the
two-dimensional
case.
M (GM r )1@2. Since !
is usually positive, the migration
A B
p
Questions?