2. Mass & Radius of Compact Objects Fastest pulsar and its stellar EOS CHENGMIN ZHANG National Astronomical Observatories Chinese Academy of Sciences, Beijing

3. Significance of Measuring Star mass and radius – Neutron or Quark  we can measure physical parameters of star, mass and radius, probe the nuclear physics and understand EOS  we can study the strong gravitational field, where Einstein GR might be tested

4. Neutron Stars ？ (Stairs 2004) (MT77) (Lattimer & Prakash 2004 ， 2006) 40+ NSs, M=1.4 M ⊙, R= 10 -30 km ? Radio pulsars, X-ray NS, binary systems

5. NS mass determined in Binary system MSP, PSR J0751+1807, M = 2.1(2) M ⊙ ？； Nice et al. 2004 2A1822-371, M>0.97+-0.24 M ⊙ ; Jonker et al 2003 ； （ 1.74 M ⊙ ， 2008 ） DNS: M=1.25M ⊙, M=1.34 M ⊙, double pulsars (2004)

6. PSR J0737-3039A/B Post-Keplerian Effects R: Mass ratio  : periastron advance  : gravitational redshift r & s: Shapiro delay P b : orbit decay (Kramer et al. 2005).. Six measured parameters – only two independent Fully consistent with general relativity (0.1%) A: 1.34 M ⊙ ; B: 1.25 M ⊙

7. Measured M-R relations Apparent Radius: R ∞ =R/(1-R s /R) 1/2 Gravitational redshift: z=(1-R s /R) -1/2 -1 Mass density: M/R 3 g=~M/R 2 1E1207.4-5209, Aql X-1 and EXO 0748-676 Rs=2GM: Schwarzschild radius No direct measure of radius !

8. Photon Spectra: Key to Measuring Radius For perfect Black Body: Observed Total Flux: F = 4  R ∞ 2  SB T ∞ 4 /d 2 Spectra are seldom black body: Neutron Stars have atmospheres ! Composition and Magnetic field shape the spectra. Other issues: Is the surface temperature and radiation isotropic ? RX J1856.5-3754 (Fred Walter’s Star !)

9. The Mass-Radius Gravitational Red-shift: observation of spectral lines (Cottam, et al 2002). QPOs indicate ISCO Exotic Stars

10. Typical twin kHz QPOs （ 24/35 ） Z: Sco x-1, van der Klis et al 2006 Separation ~300 Hz ~Spin ? Typically: Twin KHz QPO Upper ν 2 ~ 1000 (Hz) Lower ν 1 ~ 700 (Hz) Twin 21/27 sources ； ~290

11. Constrain star M_R by kHz QPOs Inner boundary to emit kHz QPO: ISCO, R > MAX M, R M<2.2 M ⊙ (1kHz/freq) R<19.5 km (1kHz/freq) M/R 3 relation known by model for twin kHz QPOs SAXJ 1808.4: M/R 3 by Burderi & King 1998

12. kHz QPOs from LMXBs: R-ISCO kHz QPO maximum frequency constrains NS equations of state Excluded Sco X-1

13. Striking case of RX J1856.5-3754 Truempet et al. 2004; Burwitz et al. 2003  Apparent radius R ∞ =16.5 km (d/117pc), Truempet 2005  True radius 14 km (1.4 M ⊙ ), stiff EOS, rule out quark star (Pons et al, 2002; Walter & Lattimer, 2002 ) This is an isolated neutron star (INS), valuable because: We can see the surface There are minimal magnetospheric complications If we can see the surface, we can determine the angular diameter The parallax gives the radius R spectral lines give the surface composition, T, and g R and g give M M/R constrains the EOS of matter at nuclear densities Gravitational light bending effect: R/M <~10 km/M ⊙ ; Ransom et al 2004

14. Einstein’s General Relativity: Perihelion precession Precession Model for KHz QPO, Stella and Vietri, 1999 ν 2 = ν kepler ν 1 = ν precession = ν 2 [1 – (1 – 3Rs/r) 1/2 ] ∆ν = ν 2 - ν 1 is not constant ISCO Saturation Relativistic precession model by Stella & Vietri 1999 M inferred from twin kHz QPOs Max frequency – ISCO

15. M/R 3 inferred from twin kHz QPOs Max frequency – Star Surface R Kepler frequency ν k = (GM/4 π 2 r 3 ) 0.5 ν k = 1850 (Hz) A X 3/2 ν 1 = ν 2 X (1- (1-X) 1/2 ) 1/2 A 2 =m/R 6 3; X=R/r, m=M/M ⊙, R 6 = R/10 6 cm Zhang 2004, AA; Li & Zhang 2005 Maximum kHz QPO occurs at R or ISCO=3Rs A> ν k /1850 (Hz) and m < 2200 (Hz)/ ν k Miller et al 1998

16. Constraining M – R by R ∞ and z 1E 1207.4-5209: R ∞ =4.6 km, Bignami et al 2004 z=0.12-0.23; Sanwal et al 2002 ? R 6 =R ∞6 /(1+z) M=f(z)R ∞6 /(1+z) F(z)=(20/3)z(1+z/2)/(1+z) 2

17. Constraining M – R by R ∞ and A~M/R 3 Aql X-1 : 9 km<R ∞ <18 km, Rutledge et al 2001 one kHz QPO: 1040 Hz; van der Klis 2006 R 6 =R ∞6 /(1+0.15(A/0.7) 2 R 2 ∞6 ) 0.5 m=AR 3 6

18. Constraining M – R by A=M/R^3 and z EXo 0728-676: z=0.35; Cottam et al 2002 One kHz QPO 695 Hz; Homan & van der Klis 2000 R 6 =1.43f 0.5 (z)(0.7/A) m=1.43f 1.5 (z)(0.7/A) f(z)=(20/3)z(1+z/2)/(1+z) 2

19. 1E1207.4-5209, Apparent radius, gravitational redshift QUARK STAR ?

20. Aql X-1, Apparent radius=14 km, single kHz QPO

21. EXO 0748-676, gravitational redshift, kHz QPO

22. Mass-Radius relations Apparent Radius: R ∞ =R/(1-R s /R) 1/2 Haensel 2001 Gravitational redshift: z=(1-R s /R) -1/2 -1 Cottam et al 2003, z=0.35 Mass density: M/R 3 (by kHz QPOs) Zhang 2004 1E1207.4-5209, Aql X-1 and EXO 0748-676 Rs=2GM: Schwarzschild radius Measuring NS Mass & Radius by kHz QPO, gravitational redshift and apparent radius

23. Measuring STAR Mass-Radius by kHz QPO, gravitational redshift and apparent radius CN1/CN2: normal neutron matter, CS1/CS2: quark star CPC: Bose-Einstein condensate of pions Zhang, Yin, Li, Xu, Zhang B, 2007 AqlX-1 ， EXO 0748-676 Samples

24. How about the Sub-millisecond Pulsar XTE J1739285, spin=1122 Hz Spin=1122 Hz Radio PSR, 716 Hz Quark Star, FAST target Cheng et al 1998, Li 1999; Xu, Qiao, Wang 2002 Horvath 2002 Harko, 2005 Zhang,..Li, 2007 More……

25. ISCO condition, m ≤ 2200 (Hz)/spin Keplerian at R, crust split

26. Zhang et al. 2006 Max kHz QPO 1330 Hz Cir X-1 difference Ratio

27. Spin Frequency - LMXBs 23 Spin sources, Av ~ 400 Hz Radio MSP ： Max Spin=716 Hz Spin frequency: Max: 1122 Hz, Kaaret et al 2007 Min: 45 Hz Villarreal & Strohmayer 2004

28. kHz QPO & spin relation

29. List of the Low-Mass X-Ray Binaries Simultaneously Detected Twin Kilohertz QPO and Spin Frequencies QPO (Hz) spin Dnu/spin 4U 160852........ 802–1099 619 1.3 4U 163653........ 971–1192 581 1.7 4U 170243......... 1055 330 3.2 4U 172834......... 582–1183 363 1.6 KS 1731260........ 1169 524 2.2 4U 191505......... 514–1055 270 1.9 XTE J1807294...... 353–587 191 1.8 SAX J1808.43658....694 401 1.7 QPO data, Belloni et al. (2005), van der Klis (2006)

30. Fastest Pulsar XTE J1739-285 spin = 1122 Hz M – R Kaaret et al. 2007 Quark Star ? Quark Star = sub-MSP ?

31. Summary THANKS Conclusions: M-R relations 1.Mass, measured 2.Radius, not measured directly 3.Spectra, MR relation 4.Redshift, M/R 5.kHz QPO, M/R^3, constraints 6.Others… Ozel 2006 Not clear: fuzzy in M-R EOS: Quark or Neutron ?

33. Saturation of kHz QPO frequency ? ISCO – Star Mass 4U1820-30, NASA Swank 2004; Miller 2004 BH/ISCO: 3 Schwarzschild radius Innermost stable circular orbit NS/Surface: star radius, hard surface