1. A Toy Model for 3:2 HFQPO Pairs in Black Hole Binaries XTE J1550-564 and GRO J1655-40 Ding-Xiong WANG( 汪定雄 ), Zhao-Ming GAN (speaker, 甘朝明 ), Chang-Yin HUANG( 黄昌印 ) and Yang LI( 李洋 ) Department of Physics, Huazhong Univ. of Sci. and Tech., Wuhan Email me: ganzhaoming@smail.hust.edu.cnganzhaoming@smail.hust.edu.cn Submitted to ApJ, 2007

2. Outline The following issues are involved Association of the 3:2 HFQPO pairs with the broad Fe K \alpha line, Association of the 3:2 HFQPO pairs with the broad Fe K \alpha line, “ Hot spots ” model for HFQPOs, “ Hot spots ” model for HFQPOs, “ Non-zero torque boundary condition ” of an accretion disk, “ Non-zero torque boundary condition ” of an accretion disk, Predicting Black-hole spin by HFQPO pairs and the broad Fe K \alpha line, Predicting Black-hole spin by HFQPO pairs and the broad Fe K \alpha line, The small change and systematic shift of the HFQPOs. The small change and systematic shift of the HFQPOs.

3. FIGURE 3. Four pairs of HFQPOs observed in three BHB systems. (Remillard et al, 2004, AIPC, 714, 13)

4. Remillard et al., 2002a, ApJ, 580, 1030 × no QPO ▲ LFQPO □ 92 Hz ○ 143Hz ■ 184 Hz ☆ 276 Hz

5. Remillard et al., 2002a, ApJ, 580, 1030 × no QPO ▲ LFQPO ☆ 276 Hz

6. FIGURE 2. Sample spectra and PDS for XTE J1550-564 illustrating the 3 states of active accretion in black hole binaries. (Remillard et al, 2004, AIPC, 714, 13) Fe K 184Hz

7. Association of HFQPOs with Fe K lines The 3:2 HFQPO pairs are not always observed simultaneously; Most detections of HFQPOs occur in the Steep Power-Law (SPL) state; Fe K \alpha lines are sometimes observed in the SPL state with HFQPO, which require a steep emissivity in the disk ( Miller 2007 ). There exists a systematic shift away from disk dominated flux with the increasing power-law flux as the HFQPO pairs shift from the higher frequency to the lower frequency;

8. Hot Spots Model for HFQPOs Kerr black hole, non-axisymmetric magnetic field; Corona: to interpret the SPL state; Boundary torque: provides the inner hot spot (thermal radiation) and the steep emissivity; Screw instability: provides the outer hot spot (non- thermal emission).

9. Since the inner and outer hotspots are produced by non-axisymmetric magnetic field and the field lines are frozen in the disk, we infer that these hotspots rotate with the Keplerian angular velocity of the disk, which refer to the higher and lower frequencies of the 3:2 HFQPO pairs ( ):

10. Non-zero Boundary Torque There are two main effects of the boundary torque: (1) The profile of radiation in the inner disk can be significantly changed and the inner hot spot move onto the inner edge of the disk; (2) If the torque is strong enough, a radial force can be exerted at the inner edge of the disk, resulting in an outward displacement of the inner edge from r_ms to r_in:

11. FIGURE 2. Curves of log(F gin /F 0 ), log(F DA /F 0 ), log(F MC /F 0 ) versus r/r in in solid, dashed and dotted lines, respectively. These curves are plotted for XTE J1550-564 with m BH = 10.8, a_*=0.7, n=9.99 and \delta=3.29*10^(-4).

12. Predicting Black Hole Spin The time scale of HFQPOs and the widely broadened Fe K lines reveal that they come from the inner most part of the accretion disk, where is ruled by the strong gravity and frame dragging of the black hole. The simultaneous observation of 3:2 HFQPOs pairs and Fe K alpha line provides serious restriction to theoretical models, however, it is also a scarce chance for us to predict the black hole spin.

13. FIGURE 3. The curves of the emissivity index \beta versus the black hole spin a_* for XTE J1550-564. The solid and dotted lines correspond to the upper and lower limits to the black hole mass, respectively. The shaded region indicates 3.95 < \beta <4.12, which corresponds to 0.7< a_*<0.998.

14. The black hole spin could be estimated, if the value range of the emissivity index can be determined by the future observations Emissivity index

15. Small Change and Systematic Shift of HFQPOs The 3:2 HFQPO pairs could have a small, but significant, change in frequency (Remillard et al., 2002a)

16. We infer that the outer hotspot is provided by the screw instability and dominated by non-thermal emission. Thus the ratio of the disk bolometric flux to the power- law flux is greater for the inner hotspot than that for the outer hotspot, and the systematic shift away from disk dominated flux with the increasing power-law flux as the HFQPO pairs shift from the higher frequency to the lower frequency based on the observations of XTE J1550-564 and GRO J1655-40 can be interpreted qualitatively within our model.

17. Thank you !