the orbital decay of a retrograde planet in a ...retrograde planets migrate inward with timescales...
TRANSCRIPT
The Orbital Decay of a Retrograde Planet in a Protoplanetary Disk
ISIMA 2010
Clément Baruteau, Jeffrey Fung
Outline
● Observations of transiting retrograde planets.– Possible explanations for retrograde orbits.
● Drag forces.– Gravitational drag & hydrodynamic drag.
● Evolutions of semi-major axis and inclination.● Simulations● Conclusions
They are out there...
● HAT-P-7b (Narita et al 2009, Winn et al 2009)● WASP-17b (Triaud et al 2010, Anderson et al 2010)● WASP-8b (Queloz et al 2010)● WASP-2b, WASP-15b (Triaud et al 2010)
Hot Jupiters, nearly cicular orbits, highly inclined
How did they get there?
● Orbital precession of a circumbinary planet.
● Takes a long time to make it retrograde.
How did they get there?
● Planet-planet scattering.● Planets could be put on
retrograde orbits before the gas disk evaporates.
The Supersonic Perturber
● A retrograde planet is moving at a very high speed with respect to the gas.
● Experiences hydrodynamic drag.
c s=Hr vk=0.05 vk
u=2 vk
M=uc s=40
The Supersonic Perturber
● Momentum exchange with gas through gravitational interaction.
● Described by Chandrasekhar's formula for dynamical friction.
● We call this the gravitational drag.
The Supersonic Perturber
● The expressions for hydrodynamic and gravitational drag force can both be written in this form:
● The difference between the two forces lies in I and σ.
F DF= I u2
I :drag coefficient , :background density :cross section ,u : relative speed
Hydrodynamic Drag
● For the hydrodynamic drag the cross section is simply the surface area of the planet: σ = π r
p2.
● The drag coefficient is dependent on the physical properties of the planet and the gas. For a sphere in a fluid of high Reynolds number (Re>104), I = 1.
● For a planet with an atmosphere...
Gravitational Drag
● The cross section in this case is computed by the gravitational focusing length, called the Bondi radius:
● For a collisionless medium, the drag coefficient for a supersonic perturber (Ostriker 99) is:
● For a collisional medium, Kim & Kim 09 did a numerical computation and got: , for M>>1
r B=G M p
u2 , =4 r B2
I=lnrmax
rmin
I=lnrmax
rmin r B
r s −0.45
Two Regimes
● If rB>r
p, then a standing bow shock is generated
ahead of the planet, and the fluid motion within the shock is sub-sonic.
– Only gravitational drag is important.● If r
B<r
p, then the shock is pressed onto the
planet surface.– Both gravitational and hydrodynamic drag
should be included.
Equations for the evolutions of a, e and i
F R=−I ∣u∣2uR
∣u∣, F=−I ∣u∣2 u
∣u∣, F z=−I ∣u∣2
u z
∣u∣
uR= GM star
a 1−e2e sin , u z=GM star
rsin i coso ,
u= GM star
a 1−e21e cosGM star
rcos i ,
=o r1AU
−1.5
e− z
2 h r 2
z= a 1−e21e cos
sin i sin o , r= a 1−e21e cos 1−sin2i sin2o
Equations for the evolutions of a, e and i
dad
= 2aGM star 1−e2 a 1−e2
1e cos 2
F R e sinF1e cos
ded
= 1GM star
a 1−e21e cos
2
F R sinF1−e2
e− 1−e2
ee2 cos di
d = 1
GM star 1−e2 a 1−e21e cos
2
F z1−e2
1e coscos o
Orbital Decay of the Planet
Orbital Decay of the Planet
Orbital Decay of the Planet
Orbital Decay of the Planet
Limitations of the Model
● Assumed the disk remains in the unperturbed state.
– Perturbation is sufficiently small such that the disk can recover within one orbital period.
● Large eccentricity cannot be taken into account.– Introduces higher order terms in the drag force.
● Global simulations are needed.
The 2D Wind Tunnel
● As a first step, 2-dimensional local simulations were done to compute the drag coefficient.
● GPU-based hydro-code written in CUDA-c.
F DF3D =I 3D4 r B
2 u2
F DF2D =I 2D 2 r B u2
Computed Drag Force● Drag Force
increases linearly with mass.
● Averaging and Fitting gives I ~ 1.
Conclusions
● Retrograde planets migrate inward with timescales < 105 years.– A highly inclined orbit can increase it by a factor of 10.
● The damping of inclination indicates that it is unlikely to have a final inclination > 50 degrees.
– Implications on when did the planet become retrograde.– Possibly inclination pumping later on.
● A global simulation could be done with a rescaling of the problem.● Hydrodynamic drag on the atmosphere of a planet.
– Also relevant for prograde gas giants on eccentric orbits.