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ith typical parameters (Σ = 10 2 g cm −2 , h/r = 0.05,
= 1) we find,
8 2. Grundlagen
τI,LR ≃ 0.5 Myr. (235)
ne concludes that there is a strong argument that
ype I migration ought to play an important role in the
ormation of giant planet cores.
The above calculation of the Lindblad torque can be
erified against hydrodynamic simulations, and is considred
to be reliable. It is, however, only part of the total
orque on a planet. What about the co-orbital torque?
sing a similar linear calculation, it is possible to estiate
the corotation torque as well. This was done by
anaka, Takeuchi & Ward (2002), who derived a total
orque,
T = −(1.36 + 0.54γ)
2 Mp rpΩp
Σpr
M∗ cs
4 pΩ2p . (236)
his formula, and its corresponding migration rate, are
ften described as representing the “standard Type I
igration rate”. One should be aware, however, that
he formula is derived under conditions (isothermality,
smooth density distribution with radius) that do not
lways hold in real disks. It is not, therefore, the comlete
answer even in linear theory, and extra caution is
equired before using it under conditions far from its doian
of validity (such as when there is a sharp density
ump in the disk).
For very low mass planets, or planets embedded in
ighly viscous disks, the standard Type I migration rate
alculated from linear theory may be reliable. For higher
ass planets and / or weaker viscosities, however, it may
e more profitable (conceptually, and possibly mathe-
atically) to consider the torque in terms of the dynamcs
of closed horseshoe orbits in the co-orbital region.
hese orbits, which are analogous to librating particle
rbits in the restricted three-body problem, are illusrated
in Figure 30. As gas in the disk executes the
-shaped turns at the ends of the horseshoe, changes
n the density of the gas exert a torque on the planet.
his way of thinking about the torque was considered
y (Ward, 1991), but largely ignored until simulations
y Paardekooper & Mellema (2006) uncovered a depen-
ence of the Type I migration rate on the thermal prop-
rties of the disk. Subsequently, many authors have studed
the co-orbital Type I torque in both isothermal (Caoli
& Masset, 2009; Paardekooper & Papaloizou, 2009)
nd non-isothermal (radiative or adiabatic) disks (Kley,
itsch & Klahr, 2009; Kley & Crida, 2008; Masset &
asoli, 2009; Paardekooper et al., 2009). Currently it
ppears as if,
• The persistence of strong co-orbital torques depends
upon whether or not they are saturated. Saturation
is possible because the region of the disk
that admits horseshoe orbits is closed and relatively
small. It cannot absorb or give an arbitrary amount
FIG. 30 The nonlinear calculation of the torque on an embedded
planet, due to co-orbital gas, is derived from consideration
of the horseshoe drag. The key point is that gas at radii
close to that of the planet executes closed horseshoe-shaped
orbits in the corotating frame. Changes in density as parcels
of gas execute the U-shaped turns result in a non-zero torque
on the planet. This torque depends on how “non-adiabatic”
the gas is: does the gas cool radiatively on the time scale
of the turn? One should also note that the region that supports
horseshoe orbits is closed. In the absence of viscosity,
this means that the co-orbital gas can only exchange a finite
amount of angular momentum with the planet, after which
the torque saturates.
Abbildung 2.2.: Skizze der Hufeisenregion. Dies ist die Region in der die Korotationsdrehmomente
wirken. Das Material am Rand der Hufreisenregion vollzieht
Hufeisenorbits (engl. horseshoe orbits), die vor allem für den Drehmomentübertrag
auf den Planeten von Bedeutung sind. Material weiter innen
oder außen bewegt sich auf den üblichen kreisförmigen Orbits (engl.
circulating orbits) (Armitage, 2007).
2.3. Kleine gravitative Störungen
of angular momentum to a planet, unless it is “connected”
to the rest of the disk via viscous stresses.
Large and persistent co-orbital torques are possible
provided that the disk is viscous enough that the
torque remains unsaturated.
Wir wollen nun also eine keplersche Scheibe mit einem störenden gravitierenden Objekt
geringer Masse betrachten. Als typisches Massenverhältnis ist das Verhältnis zwischen
Sonne und Neptun zu nennen, welches etwa q ≈ 10−4 beträgt. Da es sich nur um eine
geringe Störung handelt, lässt sich das Problem durch eine lineare Störungsanalyse untersuchen.
Sei nun Ω die Winkelgeschwindigkeit der ungestörten, keplersch rotierenden
Scheibe:
• The torque depends upon the cooling time scale
of the gas in the co-orbital region, measured relative
to the time required for the gas to execute
the horseshoe turns. Outward migration under
the combined influence of co-orbital and Lindblad
torques (which remain negative) GM⋆ may be possible in
the inner regions Ω (r) of= the disk, where the high optical
depth results in a long cooling time.
These results are all very new, and it is reasonable to
expect that further revisions to our understanding may
still occur. In particular, little work has yet been done
to address the question of how realistic turbulent flows
within the disk affect the torque and its saturation.
To summarize, Type I migration torques remain poorly
understood. The co-orbital torque is probably important,
but the mathematical relation between the linear
47
r 3 . (2.20)
Die radiale Geschwindigkeit sei Null und der Satellit befinde sich auf einem kreisförmigen
Orbit. Das Gravitationspotential Ψ ergibt sich aus dem des Zentralobjekts und dem des
Satelliten:
Ψ = Ψ⋆ + Ψs. (2.21)
Entsprechend der Scheibengeometrie bieten sich Zylinderkoordinaten (r, ϕ, z) zur Beschreibung
des Problems an. Wir beschränken uns auf das zweidimensionale Problem,