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. =H = cs r vk 0.05vk = u 2 vk M =u =40 cs ● Experiences hydrodynamic drag.

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: =  2 F DF I u

I :drag coefficient ,:background density  :cross section ,u :relative speed

● The difference between the two forces lies in I and σ.

Hydrodynamic Drag

● For the hydrodynamic drag the cross section is simply the surface area of the planet: σ = π r 2. p ● 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: G M r = p , =4r 2 B u2 B ● For a collisionless medium, the drag coefficient for a supersonic perturber (Ostriker 99) is: r I =ln max  r min ● For a collisional medium, Kim & Kim 09 did a numerical

computation and got: −0.45 r r =  max  B  I ln , for M>>1 rmin r s

Two Regimes

● If r >r , then a standing bow shock is generated B p ahead of the planet, and the fluid motion within the shock is sub-sonic. – Only gravitational drag is important. ● If r

Equations for the evolutions of a, e and i

=− ∣∣2 uR =− ∣∣2 u =− ∣∣2 u z F I u , F  I u , F I u R ∣u∣ ∣u∣ z ∣u∣ GM GM u = star e sin  , u = star sin i cos , R a1−e2 z r o

= GM star    GM star u  1 e cos  cosi , a1−e2 r − 2 1.5 − z  r 2h r =   e o 1AU a1−e2 a 1−e2 z= sinisin , r= 1−sin2isin2  1e cos o 1e cos o

Equations for the evolutions of a, e and i

 − 2 2 da = 2a  a 1 e       F e sin F  1 e cos   − 2   R d GM star 1 e 1 e cos

 − 2 2 − 2 − 2 de = 1  a 1 e    1 e − 1 e  F sin F     R  2  d GM star 1 e cos e e e cos

2 di 1 a1−e2 1−e2 =   F cos    − 2   z   o d GM star 1 e 1 e cos 1 e cos

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.

3D = 3D   2  2 F DF I 4 r B u

2D = 2D     2 F DF I 2 r B u

● GPU-based hydro-code written in CUDA-c.

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.