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

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

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

NS mass determined in Binary system MSP, PSR J , M = 2.1(2) M ⊙ ？； Nice et al A , M> M ⊙ ; Jonker et al 2003 ； （ 1.74 M ⊙ ， 2008 ） DNS: M=1.25M ⊙, M=1.34 M ⊙, double pulsars (2004)

PSR J A/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 ⊙

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 1E , Aql X-1 and EXO Rs=2GM: Schwarzschild radius No direct measure of radius !

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 J (Fred Walter’s Star !)

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

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

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 : M/R 3 by Burderi & King 1998

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

Striking case of RX J Truempet et al. 2004; Burwitz et al  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

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

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

Constraining M – R by R ∞ and z 1E : R ∞ =4.6 km, Bignami et al 2004 z= ; 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

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

Constraining M – R by A=M/R^3 and z EXo : 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

1E , Apparent radius, gravitational redshift QUARK STAR ?

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

EXO , gravitational redshift, kHz QPO

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 E , Aql X-1 and EXO Rs=2GM: Schwarzschild radius Measuring NS Mass & Radius by kHz QPO, gravitational redshift and apparent radius

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 Samples

How about the Sub-millisecond Pulsar XTE J , 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……

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

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

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

kHz QPO & spin relation

List of the Low-Mass X-Ray Binaries Simultaneously Detected Twin Kilohertz QPO and Spin Frequencies QPO (Hz) spin Dnu/spin 4U – U – U U – KS U – XTE J – SAX J QPO data, Belloni et al. (2005), van der Klis (2006)

Fastest Pulsar XTE J spin = 1122 Hz M – R Kaaret et al Quark Star ? Quark Star = sub-MSP ?

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 ?

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