遨遊天際Wondering around the Sky

江瑛貴國立清華大學

Are Planetary Systems Quantized ?

江瑛貴(Ing-Guey Jiang)國立清華大學

Outline

1. The discoveries2 The data3. The mass & period function 4. Cluster Analysis5. Our results6. The discussions & implications

The Discoveries

• 188 extra-solar planets (10 May 2006)• Methods: Doppler effect, transit, micro-len

sing, imaging• 176 exoplanets by Doppler effect• About 7 confirmed by both transit & Doppl

er effect• Wide range of mass & period & eccentricity

The Data

The (Statistical) Trend

• A deficit of high mass planets at separations smaller than 0.3 AU

• Implies a mass-period correlation• Not an observational bias (Jiang et al. 2003,

Gu et al. 2004)• Another deficit of sub-Jupiter mass planets

at separations larger than 0.5 AU ?

The Mass & Period Function

(Tabachnik & Tremaine 2002) (maximum-likelihood method) • Only 72 planets were used• Assuming a simple power-law distribution• 4 percent of solar-type stars have planets

Questions

• Is single-power law a good approximation ?• How about broken double-power law ? • There are about 176 exoplanets now.• Would the result change a lot if much more

planets are included ?• We decided to group the data first• Then, determine the mass distributions etc. f

or each group

Cluster Analysis

• To find groups in data by an algorithm• An important field in statistics• Used in a few astronomy papers• Zappala et al. (1995) identified the dynamical fami

lies from 12487 asteroids• Baggaley & Galligan 1997, Galligan 2003 used Cl

uster Analysis in meteoroid stream searches• Only Single Linkage Algorithm was used in astro.

Single Linkage Algorithm

Dendrogram

Similarity-Based Clustering Method (SCM)

• Assume there are c clusters, with c cluster centers z_i

• Define a dis-similarity function J

• Find maximums, i.e. peaks, of J by setting dJ/dz_i=0 & form a iteration algorithm

• z_i will change positions at each iteration

The Example

Our Results

• A collaboration with statistics experts: Prof. Hung, Wen-Liang & Prof. Yang, Miin-Shen

• We use the data of April 2005 of the Extrasolar Planets Catalog

• We consider those planets which projected mass, period, orbital eccentricity and semi-major axis are all available

• 143 exoplanets are in our data set

The Mass

The Period

The Mass & Period

The Period & Eccentricity

Main Results

• The mass distribution is continuous, i.e. there is no particular gap, in which planets cannot be formed

• There are two clusters in terms of period• Planetary migration did happen• Likely to have more than one migration

mechanisms: tidal & disc interactions• Eccentricity-period correlation is strong

Discussions• Our method, SCM, can determine the number and centers

of groups reasonably• It is not good to simply assume a cluster number• It can be used in other astronomical data set• To fit the data with a distribution function (as the

astronomers usually do), it would be good to do clustering analysis first

• We then can try to fit the data in each group with functions

• These functions can be more meaningfully explained by physical mechanisms

Implications on Protoplanetary Discs/Debris Discs

• In the gas-rich environment , the disc influences the planet (Trilling et al. 2002, Jiang & Yeh 2004)

• The existence of planets might also affect the disc structure

• Even after the gas is depleted, the planet might influence the structure of debris disk, the dynamics of dust grains.

• We are working on structures of debris discs etc.• It will be great to have images on gas, dust grains

etc. all together to have a complete story