Download L14 Orbital migration

Lecture 14
Orbital
migration
Klahr & Kley
Lecture Universität Heidelberg WS 11/12
Dr. C. Mordasini & PD Dr. H. Klahr
Based partially on script of Prof. W. Benz
Mentor Prof. T. Henning
Lecture 14 overview
1. Basic considerations
2. Impulse approximation
3. Gap formation and type I migration
4. Resonant torques
5. Type I migration
6. Recent results on type I migration
1. Basic considerations
Orbital migration
We have learned in the last lecture why one could expect from planet formation theory (necessity
of a 10 Mearth core formed during the disk lifetime, i.e. rapidly) that giant planets should form in a
region outside the iceline, i.e. at ~3-5 AU. The fact that the giant planets are found in our Solar
System at such a distance and further out was regarded as a good confirmation of this theory.
The detection of the first extrasolar planet by Mayor and Queloz in 1995, which was a giant
planet at an orbital distance of just 0.05 AU was therefore for many a major surprise.
ApJ, 241, 425 (October 1, 1980)
It let to the revision of the
standard picture of planet
formation (~in situ formation),
and the insight that the orbital
migration of planets represents
a key aspect of the theory
which must be included.
Ironically, migration was
discovered 15 years before the
first exoplanet by theoretical
considerations.
Basic mechanism and types
The presence of a planet orbiting the star creates a non-axisymmetric time varying gravitational
potential. The gas reacts to this perturbation in the potential by the formation of density waves.
These density waves create additional perturbations in the potential which are seen by the
planet as well. The torques originating from these perturbations change the planet’s angular
momentum and give rise to migration. We here consider migration due to the gravitational
interaction with the gaseous disk only. Migration can also occur due to the interaction with the
planetesimals disk. Orbital decay due to direct gas drag is negligible at planetary masses.
Note:
- for small mass planets the density waves
propagate through the disk
- for larger mass planets, a gap opens in
the disk
Type I migration
migration mode of small mass planets, no gap
Type II migration
surface density
migration mode of large mass planets, with gap
Simulations by P. Armitage
The movie shows the transition by ramping up the
planet mass.
Inertial and rotating frame
These hydrodynamic simulations show the development of density waves once in the inertial
and one in the co-rotating (with the planet) frame. The basic mechanism of angular momentum
exchange is that the heading density enhancement pulls the planet forward (leading to outward
migration, while the trailing density enhancement pulls the planet backwards (leading to inward
migration).
forward pull: Outwards migration
inertial frame
backward pull: Inwards migration
rotating frame
Simulations by C. Baruteau
Net torque
The net torque is the sum of all the torques. For most of the disk structures, this net torque is
such that it induces inward migration.
W. Kley
2. Impulse approximation
(30.02)
Stellar Relaxation Time
!
⇒
(∆E) = E
(30.01)
2
[Chandrasekhar 1960, Principles
of Stellar
Dynamics,Time
Chap II]
two-body encounters,
Stellar
Relaxation
T
=⇒
sin
ϕ=1
D
efine
“significant”
as
the
time
it
takes
[Ostriker
&
Davidson
1968,
Ap.J.,
151,
679]
ent, and c) close en[Chandrasekhar
1960, Principles of Stellar Dynamics, Chap II]
of its original
trajectory,
i.e.,
-range
encounters,
so
collide?& Are
interactions
between151,
stars679]
(as opposed
! Do stars ever
[Ostriker
Davidson
1968, Ap.J.,
We then assume that a) all deflections are two-bo
2
Under
these
assump=⇒ A sin
= general
1 approach
to ϕthe
system (30.02)
potential)
important?
can answer
this on a planet is the so called impulse
simple
tocollide?
compute
the We
torque
exerted
Do
stars
ever
Are
interactions
between
stars
" 1), and question
we can use
b) each
encounter is statistically independent, an
Stellar
Relaxation
Time
by calculating the time
it takes
for a star’s
orbit
to (as opposed
approximation
& Papaloizou
1979).
We
consider
thethisgravitational interaction between the
ll deflections
are two-body
encounters,
to the(Lin
general
system
potential)
important?
We
can answer
≈ v).
al
be “significantly”
perturbed
by
individual
encounters
with
other
counters
are
[Chandrasekhar
1960, Principles
ofitStellar
Dynamics,
Chap
II] toinsignificant compared to long-range
istically
independent,
and c)
close
enquestion
by
calculating
the
time
takes
for
a
star’s
orbit
planet
and
gas
flowing
past.
We
neglect
the
effect
that
we are in a corrotating frame (around
stars. To calculate
this
relaxation
time, Ap.J.,
let’s first
define
the word
[Ostriker
&
Davidson
1968,
151,
679]
compared to long-range
encounters,
so
that
during
each encounter, |∆E| " E. Under
be “significantly”
perturbed
bythrough
individual
encounters
with other
“significant”.
One
way
of
doing
this
is
total
energy:
the"sun),
and
treat
thecollide?
interaction
intime,
thebetween
twofirst
body
problem.
Here we follow the derivation in
er, |∆E|
E. Under
these
assumpstars.
To calculate
this Are
relaxation
let’s
define
the
word
Dokinetic
stars
ever
interactions
stars
(as
opposed
when
does
the
energy
exchanged
during
stellar
encounters
tions, all the deflections are small (sin ϕ " 1), a
are small
(sin ϕ " 1),
and
we
can use
Papaloizou
&
Terquem
1999
and
Alexander
2011.
“significant”.
One
way
of
doing
this
is
through
energy:
to
the
general
system
potential)
important?
We
cantotal
answer
this
equal
the
star’s
original
kinetic
energy,
i.e.,
where (vinit ≈ vfinal ≈
v). does the
the
Born
approximation,
where (vinit ≈ vfinal ≈ v)
when
kinetic
during
stellar
encounters
question
by calculating
theexchanged
time it takes
for
a star’s
orbit
to
! energy
2
(∆E)
=byEenergy,
(30.01) with other
be “significantly”
perturbed
individual
equal
theTstar’s
kinetic
i.e.,encounters
E =⇒original
n angle, ϕ is related to
the To calculate this relaxation
! time,2 let’s first define the word
stars.
we’ll define
“significant”
as
the time
it takestotal energy:
=⇒of doing
(∆E)
=E
(30.01)
! But for simplicity,
E
“significant”.
OneT
way
this
is
through
We
start
by
deriving
the
general expression for the gravitational
"
a star to losewhen
all memory
of
its
original
trajectory,
i.e.,
bv
the kinetic
exchanged
during
stellar
encounters
But for does
simplicity,
we’llenergy
define
“significant”
as
the
time
it takes
!
deflection
angle for the
case of a body of mass m, initial
! ∞
2kinetic
equal
the
star’s
original
energy,
i.e.,
TD =⇒
sin of
ϕ its
= 1original trajectory,
(30.02)i.e.,
a star to lose
all memory
1
!
=
F⊥ dt
(30.03)
relative
velocity
v and (30.01)
an impact parameter b encountering a
!
2
m
2
T
=⇒
(∆E)
=
E
v
−∞
E
M the
TD
sin ϕ =encounters,
1
(30.02)
Wedeflection
then assume
deflections
are
ounter,
angle,
related
to=⇒
the
For
a atwo-body
single
encounter,
the
deflection
angle, ϕϕ isisrelated
relatedtototh
! thatϕa)isall
For
single
encounter,
the
deflection
angle,
big
body
with
mass
M.
each encounter
statistically
independent,
and c) close
en-time it takes
arameter, b)
b, by
vBut forissimplicity,
we’ll
define “significant”
as the
initial
impact
parameter,
b,
We
then
assume
that
a)
all
deflections
are
two-body
encounters,
initial
impact
parameter,
b, by
by
counters are ainsignificant
compared
tooflong-range
encounters,
soi.e.,
star
to
lose
all
memory
its
original
trajectory,
#" #
b)each
eachencounter,
encounter|∆E|
is statistically
independent,
and c) close enduring
" The
E. !
Under
these
assump! ∞
! ∞
force
perpendicular
to the initial velocity means for
Mm
bthat
2 long-range encounters, so
1
(30.04)
counters
are
insignificant
compared
to
T
=⇒ϕ " 1),
sinand
ϕ =we1 can use
(30.02)
D(sin
2
tions, alldvthe
are
v⊥ r=
dt small
(30.03)
r=⇒
⊥ =deflectionsF⊥
that m
during
encounter,
" E. Under these assump- !
!
−∞ each
!
!
the −∞
Born approximation,
where
(vinit ≈ v|∆E|
final ≈ v).
∞
∞
∞
We then
assume
that a) are
all deflections
are
two-body
encounters,
tions,
all the
deflections
small (sin ϕ
"
1),
and
we
can
use
dv
1
dv
1
⊥⊥
b)sBorn
each approximation,
encounter is statistically
independent,
and
c) close
F
=
m
=⇒
=
FF⊥⊥dtdt (30.03
try of the encounterthe
F
=
m
=⇒
vv⊥⊥en=
dv⊥
=
(30.
where⊥(v
⊥init ≈ vfinal ≈ v).
⊥=
m
dtdt encounters, so −∞
m −∞
−∞
" counters
−∞
ds" # "
# "are
# sinsignificant compared to long-range
⇒ dt =
(30.05)
b
GMm
b each encounter, |∆E| " E. Under these assumpthat during
sin θ = Fv
=
b (30.04)
" rthe deflections
tions,
all
are small (sin ϕ " 1), and we can use
r
r 2r
From
the
of
the
encounter
From
the
geometry
ofthe
the encounter
From
the
geometry
encounter
the Born approximation,
where
(v
≈geometry
vfinal ≈ v). of
init
b
Born#approximation
rs
" #" #
" #
# "
#
"" ##
Mm
b
1
v
"
"
#
M
ds (30.06) ds
!
b
GMm
bb
"
r
v
b
GMm
v$ dt = vdt = ds =⇒ dt =
(30.05)
sinθθ=
= FF
=
(30.04
⊥=
v=FFsin
M FvF
=
(30.
2
v
⊥
!
r
r2
r
b
2
!
2GM
Impulse approximation
s
b
r
v
M
v
r
!
∞
2
!
∞
"ds/b
#" #" #
GMm
b
1
v
r
Also,M from vthe Born approximation
r
r
F⊥ = F rsin θ = Fr2
=r r2 2 r
(30.04)
r
r
r
Also,
from
theBorn
Born
approximation
Also,
from
approximation
Also,
from
thethe
Born
approximation
ds
Also, fromAlso,
the Born
from approximation
the
Born
approximation
v$ dt = vdt = ds =⇒ dt =
(30.05)
Also, from the Born approximation
v ds ds
ds
v
dt
=
vdt
=
ds
=⇒
dt
=
For small angles, we can use
Born
approximation,
where
for the
total(30.05)
velocity vinit ≈ vfinal ≈ v
ds
v$ dtthe
=
vdt
=
ds
=⇒
dt
=
(30.05)
$
ds
v
dt
=
vdt
=
ds
=⇒
dt
=
(30.05)
$
vv
v$ dt = vdtv=
ds==⇒
dt ds
= =⇒ dt = v(30.05)
vdt =
(30.05)
$ dt
v
v ds
So
v$ dt = vdt = ds =⇒ dt =
(30.05)
So
So So
# " # " v#
! ∞"
So
So1 ! ∞
GMm
1#
" # b" #
"
"
#
! ∞2
! ∞"
"
#
"
#
!
!
∞
∞
#
"
#
"
!
!
#
" GMm
# " GMm
#" b# " b1# ds1 #
! ∞
F!⊥∞
dt !=
(30.06)
v⊥ =So
"
#
2
1
∞∞ " 2 ! ∞
∞
1
2
GMm
b ds1 (30.06)
GMm
b
1 rb
1 mv⊥ −∞
m
v
F
dt
ds (30.06)
v
2 = 2 rGMm
1
1 = 1 2 F⊥ dt
Thus
=
=
⊥
⊥
0
2
2
F
dt
=
ds
(30.06)
v⊥ =
F
dt
=
ds (30.06)
v
=
F−∞
ds v (30.06)
v⊥ =⊥⊥ m −∞
⊥ m2 0m 0 r
m
r
r
r
v
⊥ dt =
2
2
m −∞ m m
m
!−∞
! ∞0r"r v r r# " vr# " v#
mr 0m
0
∞−∞
GMm
2
b
1
12
2
1/2
=
ds (30.06)
=(s r2 +
or, sinceor,vr⊥since
=
br 2=
2 1/2
2) F(s
2+
1/2
⊥b2dt
2
2 since
1/2
or,
b
)
=
(s
+
)
m
m 0
r
r
v
or,
since r or,
= (s
+
br )= r−∞
Since
since
(s= +
)1/2
or,
since
(sb22+
b2 )1/2
! ∞ ! ∞! ∞
! !∞∞ ! ∞
!
!
∞
∞
! ∞
! ∞ ! ds/b
2GM
2GM 2GM
b 2GM
2GMds/b
ds/b ds/b
2GM
b
2GM
2GM
! b∞2 b1/2
b
2GM
2
∞
2GM
ds/b
ds3/2
= ds = 2GM
v⊥since
=v⊥ =
r2GM
= (s + ds
b )=
ds
=
b ds
ds/b
v⊥v=
⊥ = or,
3/2
3/2
=
v
=
3/2
3/2 2 3/2
2
2
2
3/2
3/2
⊥
2 )3/2
2)
v
v
v
v
2
2
2
ds
=
2
2
2
v
v v v0⊥ =
v
3/2
0
0
(s
+
b
(1
+
(s/b)
)
0
0
(s
+
b
(1
+
(s/b)
) 3/2
2) )
0 0 (1
0
(s
) b22 ) 2 3/2
+
(s/b)
(s0v +v
b0 ) +(sb2 +
(1v+ (s/b)
)
(1
+
(s/b)
2
v ! ∞
! ∞0 (s + b )
0
(1 +(30.07)
(s/b)(30.07)
)
(30.07)
(30.07)
(30.07)
b
2GM
ds/b (30.07)
2GM
Letting
= s/b
s/b
ds =
=s/b
Letting
x
= s/b
Letting
x=
⊥
Letting
xx =
3/2
Defining
Letting
xvs/b
=
us(s
to2 evaluate
to(1
find
traversal
2 )3/2 velocity
Lettingvallows
x = s/b
0
0
+ b2 ) easily the vintegral
+$the
(s/b)
$∞
$∞
∞
$
!
! ∞
!
$
∞
∞
$
! ∞ ∞2GM
$$∞ $$$(30.07)
$
2GM
dx
2GM
x
2GM
dx
x
2GM
dx
2GM
x
!
$
∞
2GM
dx
2GM
x
$
∞
!
$
v
=
=
·
v⊥ =Letting
=
·
$
$
∞
$
⊥
v
=
=
·
$$
x=
2GM
2GM
⊥ s/b
v⊥ =
=
· $(1 +xx2 21/2
$
2 )2GM
1/2
3/2dx
1/2$x
3/2
2GM
dx
2
1/2
3/2
2
2
$
vb
vb
(1
+
x
vb
vb
)
3/2
2
$$
vb
vb
(1
+
x
)
0
(1
(10++x(1
x2 ))+=x ) vb= · (10 + ·x ) $$ 0
vb+ x 0)0 (1
v⊥ =
v⊥ =
2 )1/2 0$2 1/2 0
3/2
3/2
2
vb
vb
(1
+
x
2
$
vb
vb
(1 + x ) $∞
0
(1!+ 0x )(1 + x )
0
$0
∞
2GM
2GM
2GM
2GM
dx
2GM
x
2GM
$ (30.08)
(30.08)
=
(30.08)
=
=
(30.08)
=
v
=
=
·
$
⊥
vb2GM
vb
2GM
2
1/2
3/2
vb
vb
2
vb= v⊥(1
) $ (30.08)
Since
for small
/v+ x (30.08)
0
(1 + x tan
) ϕ≈ϕ
= vb deflections,
=
0 the angle
For the
(small) angle
we have
small
deflections,
ϕ ≈ ϕ = v⊥ /v and thus we finally find for
vb
vb tan
2GM
2GM
(30.08)
=
ϕ
=
(30.09)
vb 2GM
2
v b
Impulse approximation II
ϕ=
(30.09)
Impulse approximation III
We can now use our results form the previous page to calculate the momentum exchange.
For this, we now associate the velocity v of the body with mass m with the velocity difference
between a gas parcel and the planet and define for the changes in velocity:
∆v
(∆v = vgas − vp )
The change in the perpendicular component of the velocity is thus given as before by:
2GMp
|δv⊥ | =
b∆v
Note that since this velocity change occurs radially, it does not correspond to any change in
angular momentum. Since in a two body problem an encounter conserves energy, a change
in the perpendicular component also implies a change in the parallel component
.
From the conservation of energy (and geometry) we have
∆v 2 = |δv⊥ |2 + (∆v − δv|| )2
Evaluating this, and neglecting the quadratic term in
(small deflection)
�
�2
1
2GMp
δv|| �
2∆v
b∆v
h reduces to
Impulse approximation
IV
2G2 Mp2
δv|| #
.
parcel
b2 ∆v 3associated
The change of angular momentum of the gas
with
must be
balanced
by the(semi-major
change of angular
of the planet.
For a the
planet
with a semie orbital
radius
axis)momentum
of the planet
is a, then
change
in specific an
major axis a, this implies a change in specific angular momentum:
entum of the fluid element is
2G2 Mp2 a
∆j = a.δv|| = 2 3 .
b ∆v
Keplerian system, the direction of the angular momentum exchange can be readily unders
Note that gas exterior to the planet is overtaken by the planet and therefore interactions lead
exterior
to loss
the of
planet’s
feel a for
positive
torque
from
planet
(as
to a net
angular orbit
momentum
the planet
(a gain
for the
the gas)
while
thethe
gasplanet
interiorhas
is a h
al speed),
exchange
of angular
momentum
theingas
outwards
and (a
the p
overtakensobythis
the planet
and therefore
interactions
lead topushes
a net gain
angular
momentum
rds.loss
Gas
the
planet feels
the opposite
pushes inwards
by the
forinterior
the gas).to
The
interaction
is frictional.
The net effect:
directionitofismigration
thus depends
on plan
between
thetointerior
and outwards.
exterior torque.
ue, the
anddifference
causes the
planet
migrate
The net direction of migration thus dep
e difference
between
the interior
torques.
Note also that
the trajectory
prior toand
the exterior
local scattering
is assumed to be linear (corresponding
he to
total
torque
on the
planet can
be estimated
by on
integrating
Equation
5 overtothe
entire
a circular
orbit).
Subsequent
returns
of disc matter
circular orbit
are necessary
make
consider
an annulus
ofpersistent.
gas exterior
the planet
surface
density
Σ and width
the frictional
interaction
The to
scattering
itself, with
of course,
disturbs
this geometry,
so db
is an is
implicit
assumption
dissipative
other
processes
at work
to restoreΩ an
in that
the there
annulus
dm =
2πΣadb. that
If the
gas in or
this
annulus
has are
orbital
frequency
circular
for returning
trajectories.
thatannulus
disc viscosity
is able to provide
t has
Ωp , orbits
the timescale
over
which allThe
of hypothesis
the gas inisthe
will encounter
the plane
such an effect.
ure 6
2π
.
∆t =
|Ω − Ωp |
1
Richard Alexa
Impulse approximation V
The net torque will be the sum of all the torques (inside and outside) and will depend on
the exact structure of the disk.
To compute this net torque, let us integrate the single particle torque over all the gas in the
disk. Let us consider a small annulus outside the orbit of the planet at distance a. The
mass in the interval (b;b+db) is given by dm ≈ 2πaΣdb
If the planet has an orbital frequency
separated by
and the gas has
, the gas parcel suffers impulses
2π
∆t =
|Ω − Ωp |
small displacements
b<<a,
order expansion
mall For
displacements
b ! a we
cana first
approximate
|Ω − of
Ωpthe
| asangular frequencies yields:
� ! � !
roximate |Ω − Ωp | as
� dΩ
3Ωp 3Ω
p ��
!
!
dΩ
�
p
|Ω − Ωp | � � ! � b �
b p
!
|Ω − Ωp | #da
b 2a
=
b.
!
!
!
!
! dΩp !
da
2a
3Ω
p
!b =
# !!The total
b .change of the angular momentum of(7)
temporal
the planet must be the integral
!
da
2a
n therefore
calculate
the totaltransfer
torque
the planet
asparcels per unit time:
over the angular momentum
of on
all interacting
gas
"
on the planet as
∆j.dm
dJ
=−
.
"
dt
∆t
∆j.dm
=−
.
(8)
∆teliminate
! |ab = (3/2
We
can
assumingnear-Keplerian
quasi Keplerian orbits
andso
again
a first
minate the ∆v term∆vbyby
assuming
orbits,
that
∆vorder
# |Ωexpansion
p
tuting (and cancelling
lot of
we =
find
that p b.
ear-Keplerian
orbits, soathat
∆vterms),
# |Ω!p |ab
(3/2)Ω
=−
.
(8)
∆j.dm
dJ
dt = −
∆t
.
(8)
dt
∆t
eliminate the ∆v term by assuming near-Keplerian orbits, so that ∆v # |Ω!p |ab = (3/2)Ωp b.
! |ab = (3/2)Ω b.
eliminate
the
∆v
term
by
assuming
near-Keplerian
orbits,
so
that
∆v
#
|Ω
p
stituting
(and
cancelling
a
lot
of
terms),
we
find
that
p
Substituting yields
tituting (and cancelling a lot of terms)," we find that
∞ 8G2 M 2 Σa
dJ
p
"
2
=−
db .
(9)
∞ 8G M 2 Σa
dJ
dt = − 0
9Ω2p b4p
db .
(9)
2
4
dt
9Ωp b
0
integral
diverges,
but if we
specify
minimum
parameter
find parameter
min > 0 we
This integral
diverges
at the
innersome
boundary,
but ifimpact
we specify
somebminimum
impact
integral
diverges,
but
if (for
we specify
some
minimum
impact parameter bmin > 0 we find
bmin>0,
we easily
find
a constant
surface
density)
2
2
8G Mp Σa
dJ
= − 8G22M32 Σa.
(10)
dJ
dt
27Ωp bmin
p
=−
.
(10)
3
2
dt
27Ωp bmin
ractice, values of bmin between the Hill radius (for low-mass planets) and the disc scale-height
massive
planets)
a torque
which
with
thatand
computed
more
ractice,
values
bgive
the
Hill agrees
radius
(for
low-mass
planets)
the discfrom
scale-height
min between
Hence,
this of
formalism
explicitly
neglects
theapproximately
presence
of the
so-called
co-rotation
torques.
iledTherefore,
analyses.
massive
planets)
give a analysis
torque which
agreesonly
approximately
with
thatwhen
computed
the above
is applicable
in a nonlinear
regime
there isfrom
a more
This
simplifiedgap,
analysis
captures many
important
features
of the
planet-disc
interaction.
We
ledsignificant
analyses.
corresponding
to type
II migration.
Values
of bmin
are between
the Hill radius
that(for
thelow-mass
the strength
of captures
the
scales
with the features
surface
density
Σ,
so a more
massive
disc
This
simplified
analysis
important
of the planets).
planet-disc
interaction.
planets)
andtorque
the many
disc
scale-height
H (for massive
Then,
one finds
aWe
es more
rapid
planet
migration.
We scales
also
see
thatobtained
dJ/dt
scales
with
of the
planet’s
hat
the the
strength
of approximately
the torque
with
the
surface
density
Σ, square
so a more
massive
disc
torque
which
agrees
with
that
from
morethe
detailed
analyses.
s.
angularmigration.
momentum
Mpscales
, so wewith
conclude
that more
es The
moreplanet’s
rapid planet
Wescales
also linearly
see thatwith
dJ/dt
the square
of themassive
planet’s
While
there
exist
more
rigourous
to derive
net torque,
this expression
ets
migrate
more
rapidly
if the discmethods
conditions
arewith
thethe
same.
Note that
the second yields
part of
. The
planet’s
angular
momentum
scales
linearly
M
p , so we conclude that more massive
the
proper
scaling.
note
that:
sentence
is crucial.
As In
weparticular,
will
seedisc
shortly,
massiveare
planets
can modify
structure
ets
migrate
more
rapidly
if the
conditions
the same.
Note the
thatlocal
the disc
second
part of
ficantly (affecting
both
Σ will
and see
bmin
), so itmassive
is not generally
true
that more
massive
planets
sentence
is
crucial.
As
we
shortly,
planets
can
modify
the
local
disc
structure
- the torque scales with the surface density of the disk
ate more(affecting
rapidly. both Σ and b ), so it is not generally true that more massive planets
ficantly
min
the
torque
scales
with
the square of the planet mass
ate more rapidly.
J
1
Resonant
Torques
∝
- the migration time scale varies with the planet mass as τmig =
dJ/dt
Mp
Resonant
Torques
For fixed disk yields
conditions,
more massive
migrate
impulse approximation
approximately
the planets
right answer
in faster.
this case, but it is clear
Impulse approximation VI
Impulse approximation VII
The torque changes the angular momentum, the migration rate of a planet is thus
Using our earlier results for dJ/dt, we can rewrite the angular momentum exchange in terms of
the angular momentum injection rate for a given impact parameter
where the sign accounts for the fact that material outside injects angular momentum while
material inside takes away, f is an order unity constant (replacing the 8/9 in our earlier
calculation) to take into account the fact that we are in a rotating frame, and where
Here, the second term means that we exclude material closer than one disk scale height
from the torque. Clearly, |R-a| is the same as our impact parameter b.
We now calculate the total torque and thus the migration rate by integrating as before over
the disk, using as integral variable R instead of b, and taking into account that the surface
density varies as function of R:
Impulse approximation VIII
We now calculate the total torque and thus the migration rate by integrating as before over
the disk, using as integral variable R instead of b, and taking into account that the surface
density varies as function of R:
For conservation of angular momentum, we must take into account that the angular momentum
taken away from the planet is added to the disk, and vice versa.
If we describe the evolution of the disk as the one of a rotating viscous fluid, this leads to an
additional term in the master equation for the evolution of the surface density which we have
seen in earlier lectures:
viscous
evolution
planet
migration
When we have some mean to specify the viscosity, we now can solve a closed system of
equations coupling planet migration and disk evolution.
Impulse approximation IX
Lin and Papaloizou 1986 were the first to do such calculations. Planets are inserted at different
positions into a disk that is allowed to evolve viscously (outward spreading and inward accretion).
The upper plot shows a planet starting inside the radius of maximum viscous couple (or velocity
reversal). It is migrating inward. The lower planet is starting outside RMVC.
The plot on the right shows the corresponding semimajor axis of the planets
(solid lines) and of fluid elements starting at the same position (dotted) as a
function of time.
Note
-for the masses considered here, a gap is formed quickly.
-planets follow the viscous evolution of the disk (outward, inward migration).
The planet f migrates first outwards and then inwards, as it is overtaken by
RMVC.
Impulse approximation X
The plot shows the semimajor axis of planets
(solid lines) as a function of time for three
different starting location.
The left plot is for planets with a mass such that
is equal to 5 (higher planet mass), while on the
right it is equal to 80 (lower planet mass). If
B>>1, then the protoplanet’s moment of inertia is
small, so that it is essentially carried along by the
disk. For relatively small B, the protoplanet
provides a large reservoir of angular momentum,
and the evolution of the disk has little effect on
the protoplanet’s orbit.
This means that when the planet becomes more massive than the local disk, it partially
decouples. This is particularly important at small R (see also a vs b, and c vs d on in the last
figure).
3. Gap formation
Gap opening
We have seen previously that angular momentum is transported from the inner part of the
disk to the planet and from the planet to the outer part of the disk. Hence, gas inside the
planet looses angular momentum and moves inwards while gas outside gains angular
momentum and moves outwards. For this mechanism to result in the opening of a gap,
two conditions have to be met.
Condition I (thermal condition):
Hills sphere of a planet must be comparable to the disk scale height. If this is not true the
disc will be able to accrete past the planet away from the disc midplane. This can be
expressed by the condition:
rH = rp
�
Mp
3M∗
�1/3
≥H
�3
= 3h3p
Which implies a mass ratio planet/star of:
Mp
q=
≥3
M∗
�
H
r
p
Typically the disk aspect ratio is h≈0.05 and q ≥ 1.25·10-4 corresponding to M > 0.13 MJupiter.
Gap opening II
Condition II (viscous condition):
Viscous effect must not be able to close the gap. This can be expressed by the condition:
τclose ≥ τopen
In terms of torque, this condition is written
�
dJ
dt
�
LR
≥
�
dJ
dt
�
visc
Or recalling previous expressions:
8 G2 Mp2 rp Σ
2
≥
3πνΣr
p Ωp
3
2
27 9Ωp bmin
With ν = αcs H, and bmin = RH we get:
243π 2
q≥
αh
8
Typically h≈0.05, α = 10−2 so that q ≥ 2.39·10-3 corresponding to M > 2.5 MJupiter
In usual conditions, it is the viscosity criterion that determines the opening of a gap.
D
.1
Gap opening III
.05
Numerical simulations (Crida et al 2006) have led to the following approximate combined gap
opening criterion:
3 H
50
0
P =
+
≤1
1
2
3
4 RH
qR
r
Criterion for migration type:
4
2
Fig. 2. Disk eccentricity as a function of radius for theΩseveral
p a mode
R q is
R =for the q = 0.00
wherewith
the up
Reynolds
= 0.001
to q = 0.005number
at t = 2500 orbits,
ν
model at t = 3850. For the two lower curves q = 0.001 and q = 0.00
the outer edge of the computational domain lies at rmax = 2.5.
Kley & Dirksen 2006
type I migration: P > 1
Σ
1
t = 2500
1 M_jup
2 M_jup
.5
type II migration: P < 1
Fig. 1. Logarithmic plots of the surface density Σ for the relaxed state
after 2000 orbits for two different masses of the planet which is located
at r = 1.0 in dimensionless units. Top: q = 3.0 × 10−3 , and bottom: q =
5.0 × 10−3 calculated with NIRVANA. The inner disk stays circular
in both cases but the outer disk only in the lower mass case. For q =
5.0 × 10−3 it becomes clearly eccentric with some visible fine structure
in the gap. For illustration, the drawn ellipse (solid line in the lower
3 M_jup
4 M_jup
5 M_jup
0
1
2
3
4
r
Fig. 3. Azimuthally
radial profilesthe
of theplanet,
surface density
The averaged
more massive
the f
different planet masses, for the same models and times as in Fig.
the gap.
The width of wider
the gap increases
with planetary mass.
Type II migration
Once the planet is massive enough to open gap, the gas is being pushed away from the planet
and hence the torques diminish. The planet is kept in the middle of the gap, as if it were to be
closer to the inner edge, it would gain angular momentum, and it would migrate back outwards,
while the opposite effect would happen close to the outer edge. In a static disk, the planet
would also be static. As the disk is itself evolving on the viscous timescale, also the planet’s
orbit is evolving on this timescale:
τII
rp2
rp2
1 � rp �2 −1
=
=
=
Ωp
ν
αcs H
α H
where we have used the fact that the viscosity is given by ν = αcs H and the sound
speed is approximated by cs = HΩp
Note that in the case of type II migration, the migration timescale is independent of the mass
of the planet and only depends upon the mass of the star and the characteristics of the disk.
We have however seen that this simple picture is valid only if the planet is not too massive
(B>>1). One therefore distinguishes two regimes:
Disk dominated type II (B>>1):
Planet dominated type II (B<<1):
Clearly, in the planet dominated regime, migration is slower.
4. Resonant torques
Resonances
Imagine a planet in orbit about a star. The rotation frequency of the planet is given by its
Keplerian frequency,
The, there are special resonant places in the disk. Two types must be distinguished:
1) Corotation resonance
located where
Ω = Ωp
If the disc is Keplerian (i.e., if we neglect gas pressure and self-gravity), then the corotation
resonance is found at the planet’s orbital radius.
2) Lindblad resonances
located where m (Ω − Ωp ) = ±κ
2
where m is an integer, and the epicyclic frequency is κ =
In the case of a Keplerian potential, it is easy to see that:
�
2
dΩ
R
+ 4Ω2
dR
κ=Ω
�
Rg
a) for the + sign (rotation faster than planet): The inner Lindblad resonances
b) the - sign (rotation slower than planet): The outer Lindblad resonances
The actual position of these resonances can be readily obtained:
rILR = rp
�
m
m−1
�−2/3
and
rOLR = rp
�
m
m+1
�−2/3
Resonances II
A circular Keplerian disc therefore has a single corotation resonance, and a “comb” of Lindblad
resonances that “pile up” close to the planet. The innermost Lindblad resonance is found where
while the outermost is at
.
Material at these locations are particularly susceptible to excitation by the planet because of
their match in frequency.
3) Mean motion resonances
Imagine a second planet orbiting at
2π
Ωp
2π
Ωq
Ωq
Pp
n
=
=
Pq
m
the planets are said to be in resonance (n:m) if
where n,m are small integers
Summing resonant torques
We have studied above the impulse approximation to have a rough estimate of planetary
migration rates. A full analysis instead considers the evolution of linear perturbations in a fluid
disc, and was first applied to planet-disc interactions by Goldreich & Tremaine (1979).
The first step is to linearize the hydrodynamic equations, and to decompose the perturbation
to the Keplerian potential due to the planet into Fourier modes. For example, the potential of
the planet is expressed in terms of azimuthally periodic components characterized by m, the
inverse wave number.
Φm (r, ϕ, t) = Φm (r) cos[m(φ − Ωp t)]
The next step is to compute the response of the disc to the perturbations, and from this it is
possible to calculate the total torque on the planet by summing up the torques originating at
each resonance.
Tanaka et al 2002
Summing resonant torques II
The important result of such complex calculations is that the total angular momentum exchange
between disc and planet can be expressed as the sum of the torques exerted at discrete
resonances in the disc.
These resonances correspond to the points in the disc where the planet excites waves, which
take the form of spiral density waves.
The importance of the resonances can be understood by the following argument: the torques
at non-resonant locations in the disc do not interfere constructively, and consequently cancel
out when averaged over the orbit.
Using such an approach, Goldreich & Tremaine (1979) derived an expression for the torque at
the Lindblad resonance:
Γm = mπ
2
�
�
Σm
dΦm
2Ω
r
+
Φm
rdD/dr
dr
Ω − Ωp
with D(r) = κ(r)2 − m(Ωp − Ω(r))
��
r=rL
Summing resonant torques III
Since then, several groups performed similar calculations. For example, Tanaka et al (2002)
studied the three-dimensional interaction between a planet and a three dimensional isothermal
disk. Assuming that the perturbation are small enough, they used linear theory to derive the
relevant torques. After some significant calculations, they obtain the following expression for
the torque exerted on the planet:
�
M p rp Ω p
M∗ c s
r
r0
�−α
Γ = −(1.364 + 0.541α)
Here
�2
Σp rp4 Ω4p
α is defined by the exponent of the assumed power law surface density of the disk
Σ(r) = Σ0
�
The corresponding inward migration timescale is given by:
τmig = (2.7 + 1.1α)
−1 M∗
M∗ −1
1
Ωp ∝
2
Mp Σ p r p
Mp
5. Type I migration
Type I migration
Putting the results from the studies like Goldreich & Tremaine or Tanaka et al. together, the
migration is determined by the net torque exerted onto the planet. This torque can be written:
�
�
ΓLR + ΓCR
Γtot =
ΓLR +
ILR
OLR
This torque can be expressed in a general way as:
Γtot
1
= (C0 + C1 α + C2 β)Γ0
γ
where the generic torque is given by:
Γ0 =
� q �2
h
Σ2p rp2 Ω2p
Here C0 , C1 , and C2 are constants and α, β are the exponents of the assumed power-law
distribution of the density and temperature respectively. These constants can be derived
semi-analytically or from numerical simulations.
The conservation of angular momentum implies:
dJp
= Γtot
dt
The angular momentum of the planet is given by:
Jp = Mp (GM∗ rp )1/2
Type I migration cont.
From which we obtain the migration velocity:
drp
Γtot
= −2rp
dt
Jp
Depending upon the sign of the torque (i.e. upon the signs of the Ci and the slopes of the
density and temperature distribution), the migration can proceed inwards or outward.
For an isothermal disk with a MMSN profile (α =1.5) migration is inwards and rapid.
Migration timescales
The plot shows the migration timescale as a function of planet mass and alpha viscosity
parameter, in a disk about twice as massive as the MMSN. We note that before the transition
to type II sets in, the migration timescale in type I are very short, ~104 yrs. Afterwards,
migration timescale increase by 1-2 orders of magnitude from type I to type II.
For type I this means: Planets seem
to migrate so fast that they should
all fall into the star within the lifetime
of the disk (unless they grow
extremely rapid)!
These very short migration timescales
represent another major issue in
modern planet formation theory.
disk lifetimes
simple linear theory for isothermal
disks cannot be the final word!
Ward 1997
6. Recent results on type I
migration
Updated type I migration rates
The puzzling results of extremely fast type I migration timescales have triggered numerous
studies trying to understand which processes could slow down type I migration so that it
planet growth again becomes possible. Several possible processes have been put forwards of
which we only address here two:
1) Random walk migration in turbulent disks
There has been strong interest (e.g. Nelson & Papaloizou 2004, Uribe et al. 2011) in studying
planet-disk interactions in turbulent disks, where the turbulence is magnetically generated by
the magneto-rotational instability (MRI), instead of the laminar disks which we have used in
the above analysis. Such laminar disks are strictly speaking not self-consistent, as they
assume an abnormal viscosity (thought to be due to turbulence) without actually taking the
consequence of turbulent motions into account.
More realistically, angular momentum transport itself derives from turbulence, which is
accompanied by a spatially and temporally varying pattern of density fluctuations in the
protoplanetary disk. These fluctuations will exert random torques on planets of any mass
embedded within the disk.
In such turbulent disks, it is found that for low mass planets, Type I migration is no longer
effective due to large fluctuations in the torque. The fluctuations in the torque
created by the perturbations in the density can be larger than the mean torque expected for
standard Type I migration in a laminar disk.
Updated type I migration rates II
Under these conditions, low mass planets undergo rather a random walk like migration. Only
for planets more massive than ~30M⊕ = 0.1MJup the stochastic torques are found to
converge to the standard type I migration torques after long-time averaging.
Uribe et al. 2011
The plots show the logarithm of the disk density in the mid plane (top row) and in an
azimuthal cut at the position of the planet (bottom row) for q = 10−5 (~3 Mearth), q = 10−4 (~30
Mearth) and q = 10−3(~1 MJup). The left plot shows no significant perturbation of the density by
the planet, and no spiral arms are seen, indicating that random torques are dominating.
Non-isothermal migration
2) Migration in non-isothermal disk
Crida et al. 2006; Baruteau & Masset 2008; Casoli & Masset 2009; Pardekooper et al. 2010; Baruteau & Lin 2010
outward
inward
Kley & Crida 2009
Another important (and not
justified) assumption made in the
derivation of the classical type I
torque is that the gas around the
planet acts isothermally. In other
words, its cooling time is vanishing.
Radiation hydrodynamic
simulations treating correctly the
energy transport in the disk find
that below a certain threshold
mass, migration can be directed
outwards, due to a different density
distribution around the planet.
Thermodynamics of the disk is essential
Horseshoe orbits
A central role for the torque in non-isothermal disk is due to gas moving along so called
horseshoe orbits.
1) starting at point A, the body is moving towards
the planet and is being accelerated and this put
on higher and higher orbits.
2) eventually, at point C, the body is moving
sufficiently slower than the planet that it starts
lagging behind.
3) at point D, the planet is catching up with the
body and is pulling it back therefore reducing its
angular momentum. The body eventually moves
onto a lower orbit.
4) since the body is on a lower orbit, it will
eventually catch up with the planet and the cycle
can restart.
width of horseshoe orbit:
xs = 1.16Rp
�
q
H/Rp
libration time:
τlib
8πRp
−1/2
=
∝ Mp
3Ωp xs
Thermodynamics matter
C. Baruteau
Entropy
Perturbed density
1.04
1.02
1.02
1.00
-3
-2
-1
0
1
ϕ − ϕp
2
r/rp
1.04
1.00
0.98
0.98
0.96
0.96
3
-3
-2
simulation for 1/2 libration period
-1
0
1
ϕ − ϕp
2
The exchange of fluid elements lead to an overdensity at smaller radii. This translates into
a increased torque pushing the planet forwards and thus outwards.
For this mechanism to work, the fluid has to remain adiabatic during the exchange
process. In other words: τcool >> τu−turn
τcool
ΣcV T
ΣcV T
≈
=
4
Q
2σTef
f
τu−turn ≈ 1.16
�
h3 64
γq 9Ωp
3
Thermodynamics matter II
Unfortunately, after some time, the situation becomes much less clear and the outward
directed corotation torques begins to saturate (vanish)...
C. Baruteau
Perturbed entropy
-3
-2
-1
0
1
ϕ − ϕp
2
1.04
1.04
1.02
1.02
1.00
1.00
0.98
0.98
0.96
0.96
3
-3
-2
-1
simulation for 3 libration periods
0
1
ϕ − ϕp
2
3
In other words, unless the viscosity re-establishes the original entropy profile, the outward
migration will not last.The condition for a sustainable outward migration is therefore given
by: τlib >> τvisc
τlib
8πrp
=
3Ωp xs
τvisc
(2xs )2
=
ν
r/rp
Perturbed density
τlib
Cooling?
τcool � τU−turn
τcool � τU−turn
Isotherm.
Inward
Adia.
Outward
(outer disk)
τvisc
Subtypes of type I
By comparing the relevant
timescales, different
subtypes of type I migration
can be identified.
τu-turn
(inner disk)
Star
τcool
Saturation?
Sat.
Inward
(High Mplanet)
Unsat.
Outward
(Low Mplanet)
Dittkrist et al. (2011, in prep)
Corotation region: horseshoe orbits
w. entropy gradient: needed for
outward migration
Type I convergence zones
Time [yrs]
An important consequence of nonisothermal migration is that it leads to the
existence of convergence zones. These
are zero torque locations in which planets
get trapped.
out
out
Dittkrist et al. in prep
in
in
The exact location of convergence will
depend upon the detailed structure of the
disk (temperature and surface density).
The location of the convergence zone
itself moves inward on a viscous
timescale.This means that despite being
in the type I regime, the planets will move
inwards on a much slower viscous
timescale, as in type II.
Semimajor axis [AU]
Figure 4.2: The top plot shows the torque factor Cadia (a) for the adiabatic regime at
different
times indicated
in the
plotthese
for thezones
same disc
ERROR
IN massive
THE Y
It is tempting
to think
that
are as
thebefore.
places
to grow
AXIS LABEL. The lower panel illustrates again regions of outward and inward migration.
concentrate many growing protoplanets.
Green symbols show negative and red positive values of Cadia . Additionally, the location
of the two important convergence zones are shown.
planets, as they might
Questions?