1. Eccentricity evolution of giant planet orbits due to circumstellar disk torques
Althea V. Moorhead, University of Michigan
Fred C. Adams, University of Michigan
This project focuses on how the orbital eccentricity of a Jupiter-sized planet evolves due to the planet’s interaction with a circumstellar disk. This situation is known as Type II migration.

2. Introduction: Type II Migration
Resonances between planet and disk material exert torques
Provides explanation of small a observed
We find that disk-induced eccentricity evolution is a complex function of eccentricity, producing both damping and excitation.
Type II migration is differentiated from Type I migration in that the planet is massive enough to clear a gap in disk material in the vicinity of its orbit, instead of being embedded within disk material.
Wherever a ring of material in the remainder of the disk orbits with a period that is a rational multiple of the planets orbital period, a resonance occurs. The planet and the material at this resonance then exert torques on each other.
The torques on the planet force the planet inward, providing a possible explanation of how so many observed giant planet are currently orbiting well within where they were most likely formed.
The effect on eccentricity, on the other hand, is debated, with some saying it is always damped, and others saying it can be excited. We say that e-dot is a complicated function that attains both positive and negative values.

3. Introduction: Observations
See Moorhead & Adams 2005

4. Methods: Disk Properties
Mdisk = 0.05 M
R < r < 30 AU.
Surface density and temperature follow a power law.
 =
Flat, non-self-gravitating disk.
Point out disk properties one by one.
Mass is comparable? to MMSN.
Typical ranges in p, q.
Typical values of  range from .01 to .0001
These choices of parameters result in a disk that is flat, or cold, (h/r << 1) and is non-self-gravitating.

5. Methods: Inclusion of Torques Goldreich & Tremaine 1980
Lindblad resonances
corotation resonances
Eccentricity evolution due to torques
Now I would like to discuss the resonant torques I mentioned in more detail.
We obtain these torque formulas from expanding the planet’s and disk’s perturbations of the overall gravitational potential in cosine series and taking the overlapping terms. These terms correspond to resonances.
The phi term that appears in these torque functions is the (m,l) term of the cosine expansion of the disturbing function. It is a 2-D integral of an oscillatory function. To avoid time-expensive calculation of phi, there are a couple of approximations people have used: the small e approximation and the large m approximation.

6. Methods: Approximations
If the planetary system in question is like ours, with very circular orbits, then you have reason to expand the phi function to first order in e. Here, I compare the function and the approximation for one particular value of (m,l). We find that the phi function deviates widely from the linear approximation for eccentricity larger than 0.3.

7. Methods: Approximations Goldreich & Sari 2003
Lindblad resonances
corotation resonances
Large m approximation
Torques are stronger for large m.
Torques have similar dependence on Pl,m for large m.
However, only small planets attain large m.
If you wish to avoid making the small e approximation, there is also the large m approximation. In the large m limit, all torques have the same dependence on phi, and so you can then look at the degree to which they cancel or add. However, planets of 1-10 jupiter masses do not allow for a large number of resonances, so the approximation is not valid.

8. Results: Example Surface Density Profile
Our method is this: we choose a gap structure for our surface density profile. Here we turn to Bate for a starting point. The shape of the surface density profile affects the strengths of the different torque types and the width of the gap puts an upper limit on m. We use Mathematica to calculate the exact phi function, and then add up all contributing torques.
Bate et al assumes p = 1/2.
We approximate Bate’s gap.

9. Results: Damping and Excitation Occur
Here is the end result. In the peaks and troughs of this function we recognize the features of the phi’s. Notice that for small eccentricity, the planet consistently undergoes eccentricity damping. For larger eccentricities, the evolution quickly changes from positive to negative, depending on the exact value of the planet’s orbital eccentricity.
This plot is not extremely sensitive to the exact gap shape. As the gap width changes, peaks will appear and disappear. Beyond that, the value and derivative of surface density at different points affects the relative heights of the features.
If you don’t like this plot because it assumes a sharp cut off at the edges of the gap, I present to you a second plot...

10. Results: A partially cleared gap
Explain different regions

11. Summary Both excitation and damping occur
Moderately sensitive to gap shape
High e behavior consistent with lack of planets in very high e orbits
Can be incorporated into simulations

12. Supplement: Corotation Damping