### Tidal Dissipation in Extrasolar Planets

### Fernando Gabriel Pena

Doctor of Philosophy 2010

Graduate Department of Astronomy and Astrophysics, University of Toronto
Many known extra-solar giant planets lie close to their host stars. Around 60 have
their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast
majority of these close-in planets have low eccentricity orbits. This suggests that their
orbits have been circularized likely due to tidal dissipation inside the planets.

These exoplanets share with our own Jupiter at least one trait in common: when they
are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal “quality
factor”, Q, of order 105. This parameter is the ratio between the energy in the tide and
the energy dissipated per period. To explain this, a possible solution is resonantly forced
internal oscillation. If the frequency of the tidal forcing happens to land on that of an
internal eigenmode, this mode can be resonantly excited to a very large amplitude. The
damping of such a mode inside the planet may explain the observed Q value.

The only normal modes that fall in the frequency range of the tidal forcing (∼ few
days) are inertial modes, modes restored by the Coriolis force.

We present a new numerical technique to solve for inertial modes in a convective,
rotating sphere. This technique combines the use of an ellipsoidal coordinate system
with a pseudo-spectral method to solve the partial differential equation that governs
the inertial oscillations. We show that, this technique produces highly accurate solutions
when the density profile is smooth. In particular, the lines of nodes are roughly parallel to
the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate
the resultant tidal dissipation for giant planets, and find that turbulent dissipation of
inertial modes in planets with smooth density profiles do not give rise to dissipation as
strong as the one observed. We also study inertial modes in density profiles that exhibit
discontinuities, as some recent models of Jupiter show. We found that, in this case, our
method could not produce convergent solutions for the inertial modes.

Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by
observing waves in its rings that may be excited by inertial modes inside Saturn.