1. 遨遊天際 Wondering around the Sky 江瑛貴 國立清華大學

2. Are Planetary Systems Quantized ? 江瑛貴 (Ing-Guey Jiang) 國立清華大學

3. Outline 1.The discoveries 2The data 3.The mass & period function 4.Cluster Analysis 5.Our results 6.The discussions & implications

4. The Discoveries 188 extra-solar planets (10 May 2006) Methods: Doppler effect, transit, micro- lensing, imaging 176 exoplanets by Doppler effect About 7 confirmed by both transit & Doppler effect Wide range of mass & period & eccentricity

7. The Data

9. 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 ?

10. 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

11. 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. for each group

12. 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 families from 12487 asteroids Baggaley & Galligan 1997, Galligan 2003 used Cluster Analysis in meteoroid stream searches Only Single Linkage Algorithm was used in astro.

13. Single Linkage Algorithm

14. Dendrogram

15. 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

16. The Example

20. 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

21. The Mass

23. The Period

25. The Mass & Period

27. The Period & Eccentricity

29. 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

30. 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

31. 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