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.