Li-Chin Yeh

On the Chermnykh-Like Problems: II. The Equilibrium Points

Motivated by Papadakis (2005a, b), we study a Chermnykh-like problem, in which an additional gravitational potential from the belt is included. In addition to the usual five equilibrium points (three collinear and two triangular points), there are some new equilibrium points for this system. We studied the conditions for the existence of these new equilibrium points both analytically and numerically.

On the fate of close-in extrasolar planets

It has been shown from the current observational data that there is a possible mass-period correlation for extrasolar planets, and this correlation is, in fact, related to the absence of massive close-in planets, which are strongly influenced by tidal interaction with the central star. We confirm that the model in Patzold & Rauer is a good approximation for the explanation of the absence of massive close-in planets. We thus further determine the minimum possible semimajor axis for these planets to be detected during their lifetime and also study their migration timescale at different semimajor axes by the calculations of tidal interaction. We conclude that the mass-period correlation at the time when these planets were just formed was less tight than it is now observed if these orbital migrations are taken into account.

The drag-induced resonant capture for Kuiper Belt objects

It has been an interesting question as to why one-third of Kuiper Belt objects (KBOs) are trapped into the 3 : 2 resonance but, in contrast, only a few KBOs are claimed to be associated with the 2 : 1 resonance. In a model proposed by Zhou et al., the stochastic outward migration of the Neptune could reduce the number of particles in the 2 : 1 resonance, and thus the objects in the 3 : 2 resonance become more distinct. As a complementary study, we investigate the effect of protostellar discs on the resonance capture. Our results show that the gaseous drag of a protostellar disc can trap KBOs into the 3 : 2 resonance rather easily. In addition, no objects are captured into the 2 : 1 resonance in our simulation.

DYNAMICAL EFFECTS FROM ASTEROID BELTS FOR PLANETARY SYSTEMS

The orbital evolution and stability of planetary systems with interaction from the belts is studied using the standard phase-plane analysis. In addition to the fixed point which corresponds to the Keplerian orbit, there are other fixed points around the inner and outer edges of the belt. Our results show that for the planets, the probability to move stably around the inner edge is larger than the one to move around the outer edge. It is also interesting that there is a limit cycle of semi-attractor for a particular case. Applying our results to the Solar System, we find that our results could provide a natural mechanism to do the orbit rearrangement for the larger Kuiper Belt Objects and thus successfully explain the absence of these objects beyond 50 AU.

Data analysis on the extrasolar planets using robust clustering

We use both the conventional and more recently developed methods of cluster analysis to study the data of extra-solar planets. Using the data set with planetary mass M, orbital period P, and orbital eccentricity e, we investigate the possible clustering in the ln M, ln P, ln P-ln M, e, and ln P-e spaces. There are two main implications: (1) mass distribution is continuous and (2) orbital population could be classified into three clusters, which correspond to the exoplanets in the regimes of tidal, on-going tidal and disc interaction, respectively.

Data Analysis on the Extra-solar Planets Using Robust Clustering

We use both the conventional and more recently developed methods of cluster analysis to study the data of extra-solar planets. Using the data set with planetary mass M, orbital period P, and orbital eccentricity e, we investigate the possible clustering in the ln M, ln P, ln P-ln M, e, and ln P-e spaces. There are two main implications: (1) mass distribution is continuous and (2) orbital population could be classified into three clusters, which correspond to the exoplanets in the regimes of tidal, on-going tidal and disc interaction, respectively.

On the Mass-Period Distributions and Correlations of Extrasolar Planets

In addition to fitting the data of 233 extra-solar planets with power laws, we construct a correlated mass-period distribution function of extrasolar planets, as the first time in this field. The algorithm to generate a pair of positively correlated beta-distributed random variables is introduced and used for the construction of correlated distribution functions. We investigate the mass-period correlations of extrasolar planets both in the linear and logarithm spaces, determine the confidence intervals of the correlation coefficients, and confirm that there is a positive mass-period correlation for the extrasolar planets. In addition to the paucity of massive close-in planets, which makes the main contribution on this correlation, there are other fine structures for the data in the mass-period plane.

A Possible Correlation between the Gaseous Drag Strength and Resonant Planetesimals in Planetary Systems

We study the migration and resonant capture of planetesimals in a planetary system consisting of a gaseous disk analogous to the primordial solar nebula and a Neptune-like planet. Using a simple treatment of the drag force, we find that planetesimals are mainly trapped in the 3 : 2 and 2 : 1 resonances and that the resonant populations are correlated with the gaseous drag strength in a sense that the 3 : 2 resonant population increases with the stronger gaseous drag, but the 2 : 1 resonant population does not. Since planetesimals can lead to the formation of larger bodies similar to asteroids and Kuiper Belt objects, the gaseous drag can play an important role in the configuration of a planetary system.

Orbital Evolution of Scattered Planets

A simple dynamical model is employed to study the possible orbital evolution of scattered planets, and phase plane analysis is used to classify the parameter space and solutions. Our results reconfirm that there is always an increase in eccentricity when the planet was scattered to migrate outward when the initial eccentricity is zero. Applying our study on the solar system and considering the existence of the Kuiper Belt, this conclusion implies that Neptune was dynamically coupled with the Kuiper Belt in the early phase of the solar system.

ON THE FUNDAMENTAL MASS-PERIOD FUNCTIONS OF EXTRASOLAR PLANETS

Employing a catalog of 175 extrasolar planets (exoplanets) detected by the Doppler-shift method, we constructed the independent and coupled mass-period functions. It is the first time in this field that the selection effect is considered in the coupled mass-period functions. Our results are consistent with those of Tabachnik and Tremaine in 2002, with the major difference that we obtain a flatter mass function but a steeper period function. Moreover, our coupled mass-period functions show that about 2.5% of stars would have a planet with mass between Earth Mass and Neptune Mass, and about 3% of stars would have a planet with mass between Neptune Mass and Jupiter Mass.