The effect of planetary migration on the corotation resonance

G. I. Ogilvie 1 2 S. H. Lubow 0 1 0 Space Telescope Science Institute , 3700 San Martin Drive, Baltimore, MD 21218 , USA 1 Institute of Astronomy, University of Cambridge , Madingley Road, Cambridge CB3 0HA 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences , Wilberforce Road, Cambridge CB3 0WA A B S T R A C T The migration of a planet through a gaseous disc causes the locations of their resonant interactions to drift and can alter the torques exerted between the planet and the disc. We analyse the time-dependent dynamics of a non-coorbital corotation resonance under these circumstances. The ratio of the resonant torque in a steady state to the value given by Goldreich & Tremaine depends essentially on two dimensionless quantities: a dimensionless turbulent diffusion time-scale and a dimensionless radial drift speed. The dimensionless diffusion timescale is a characteristic ratio of the time-scale of turbulent viscous diffusion across the librating region of the resonance to the time-scale of libration; in the absence of migration, this parameter alone determines the degree of saturation of the resonance. The dimensionless radial drift speed is the characteristic ratio of the drift speed of the resonance to the radial velocity in the librating region; this parameter determines the shape of the streamlines. When the drift speed is comparable to the libration speed and the viscosity is small, the torque can become much larger than the unsaturated value in the absence of migration, but is still proportional to the large-scale vortensity gradient in the disc. Fluid that is trapped in the resonance and drifts with it acquires a vortensity anomaly relative to its surroundings. If the anomaly is limited by viscous diffusion in a steady state, the resulting torque is inversely proportional to the viscosity, although a long time may be required to achieve this state. A further, viscosity-independent, contribution to the torque comes from fluid that streams through the resonant region. In other cases, torque oscillations occur before the steady value is achieved. We discuss the significance of these results for the evolution of eccentricity in protoplanetary systems. We also describe the possible application of these findings to the coorbital region and the concept of runaway (or type III) migration. 1 I N T R O D U C T I O N 2 I N C L U S I O N O F T I M E - D E P E N D E N C E as in equation (17) of Paper I, with (r , , T ), 2 p(T ) dT , r rc(T ) , Using the chain rule we find 2 t + T a = 2 b = 2r Tc = tcTGT, where tc is dimensionless, and m2 2 d TGT = 2(d / dr ) dr B is the torque formula of GT79. 4 A N A LY S I S I N R E A L S PAC E 4.1 Interpretation as a vortensity equation The ratio As in Paper I, we consider a single potential component of the form 3 R E D U C T I O N T O A D I M E N S I O N L E S S F O R M 4.2 Streamlines of the dominant motion 1 2 a cos d + 2 x , 4.3 Behaviour of the vortensity perturbation 1 Q r drc d Q Q u dr 4.4 Estimation of the torque in the low-viscosity limit Tc = r u dr Tc x 2 d A X 2 = cos cos s + v( s) 4.5 Interpretation of the torques tc = G1(0) G1(0). 6 M E T H O D S O F N U M E R I C A L S O L U T I O N 7 N U M E R I C A L R E S U LT S 8 A P P L I C AT I O N T O E C C E N T R I C R E S O N A N C E S m p = 0.7006Cmm5/3eq2/3 r 4/3 , (53) (54) 9 T E N TAT I V E A P P L I C AT I O N T O T H E C O O R B I TA L R E G I O N 9.1 Approximate flow in the coorbital region L1, L2: x = = 0 L3: x = 0, u = 2qr sin 1 9.2 Stability of migration with a radial velocity-dependent torque 1 0 S U M M A R Y A N D D I S C U S S I O N AC K N OW L E D G M E N T S R E F E R E N C E S Artymowicz P., 2004, KITP Conference on Planet Formation, http://online.kitp.ucsb.edu/online/planetfc04 Balmforth N. J., Korycansky D. G., 2001, MNRAS, 326, 833 DAngelo G., Bate M. R., Lubow S. H., 2005, MNRAS, 358, 316 We consider the advectiondiffusion equation + u Q 2 Q = S, Q = Q0(, ) + Q1(, ) + Q0 = S. 0 = on each streamline in C. Instead, the solution is of the form Q0 Q1 2 Q1 = S, 2 Q0 = 0. I = I = I1 = Q (u Q 2 Q ) d A I = 1 I1 + I0 + , = Q S d A | Q1|2 d A, A() A() The divergence theorem implies 2 Q1 d A = 2 Q1 d A = Q1 n ds = ( Q1) Q0 d A = Q12 Q0 d A = Q1 This paper has been typeset from a TEX/LATEX file prepared by the author. (...truncated)