遨遊天際 Wondering around the Sky

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 • The discoveries • The data • The mass & period function • Cluster Analysis • Our results • 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