kHz QPOs of LMXBs Constrains on Pulsar Parameters Chengmin Zhang & Hongxing Yin National Astronomical Observatories, Beijing

OUTLINE OF TALK KHz Quasi Periodic Oscillation (QPO) in NS/LMXB Millisecond accreting-powered X-ray Pulsar Type-I X-ray Burst Oscillation Other QPOs of NS & BH Theoretical Mechanisms---kHz QPOs Constrains on Mass, Radius, Magnetic of compact object

RXTE Target A/Periodic, transient, and burst in X-ray binaries, masses, orbital, matter exchange nuclear matter in compact objects, M-R relation, magnetic field Behavior of matter into/onto a NS/BH Strong Gravity of GR near a NS/BH Mechanisms causing X-ray emission Rossi X-ray Timing Explorer (RXTE): NASA, Launched on Dec. 30, 1995

QPO frequencies by RXTE 1996—2007 kHz QPOs (27) van der Klis 2006; Belloni et al. 2005; Zhang et al Spin frequency - Burst oscillation (23); Hz Villarreal & Strohmayer. 2004; Strohmayer & Bildsten 2006; Kaaret 2007 HBO, ~15-70 Hz, van der Klis 2006 FBO/NBO, ~5-20 Hz, Yu et al 2003; van der Klis 2006 Others.

Typical twin kHz QPOs （ 21/27 ） Z: Sco x-1, van der Klis et al 1996; Atoll: 4U Strohmayer et al 1996 Separation ~300 Hz ~Spin ? Typically: Twin KHz QPO Upper ν 2 ~ 1000 (Hz) Lower ν 1 ~ 700 (Hz) Twin 21/27 sources ； ~290

kHz QPOs in Atoll and Z Sources -- CCD Accretion rate direction ~Eddington Accretion ~1% Eddington Accretion

QPO v.s. Accretion rate relation SCO X-1, Van der Klis, 2006 QPO frequency increases with the accretion rate kHz QPO profile; Mendez 2005

Max ： ν max =1329 Hz, van Straaten & van der klis 2000 min: ~200 Hz KHz QPO Data ， Atoll and Z sources Distribution of kHz QPOs ： QPO (Atoll) ~ QPO （ Z ） Zhang et al 2006 MNRAS;

Table: kHz QPO, spin 8

separation of twin kHz QPOs = const? Beat ? Strohmayer et al 1996; Miller, Lamb & Psaltis 1998;

Saturation of kHz QPO frequency ? 4U , NASA W. Zhang et al, 1998 Kaaret, et al 1999 Swank 2004; Miller 2004 BH/ISCO: 3 Schwarzschild radius Innermost stable circular orbit NS/Surface: star radius, hard surface

Parallel Line Phenomenon kHz QPO － luminosity relation Similarity/Homogeneous ? Among the different sources, same source at the different time

kHz QPO v.s. Count rate kHz QPO corresponds to the position in CCD, to accretion rate Mdot; QPO ~ Mdot

Zhang ， et al ， MNRAS kHz QPO Distribution : separation /ratio is not a constant ν 1 = ~700. (Hz)(ν 2 /1000Hz) b b ~ 1.6 Atoll Source 4U1728 b ~ 1.8 Z Source Sco X-1 Cir X-1 difference Ratio

AMXP: XTE , SAXJ : special cases 1.5 shift XTE , Linares et al 2006; Zhang et al 2006 SAXJ : Wijnands & van der Klis 2003

Twin kHz QPO distribution

Twin kHz QPO distribution; ratio

Accreting X-ray millisecond pulsar --- SAX J (8 AXMPs); 401 Hz (2.49 ms) Wijnands and van der Klis, 1998 Nat; Wijnands et al 2003 Nat

Type-I X-ray Burst frequency 17 burst sources, van der Klis 2006; Strohmayer and Bildsten U , (363 Hz) Strohmayer et al 1996 ApJ Hz Hz

SAXJ ， Twin kHz QPOs ： 700 Hz, 500 Hz ； Burst/spin: 401 Hz ； Wijnands et al 2003 ， Nature Burst frequency = spin frequency XTE , twin kHz QPOs, 191 Hz, Linares, et al 2006 ; Zhang ， et al, 2006 kHz QPO separation and spin relation Slow rotator; separation/spin ~ 1 Fast rotator; separation/spin ~ 0.5 Linares & van der Klis 2007

Spin Frequency Distribution =23 Spin sources Radio MSP ： Max Spin=716 Hz Yin, Zhang, Zhao et al 2007 AA Spin frequency: Max: 1122 Hz, Kaaret et al 2007 ApJ Min: 45 Hz Villarreal & Strohmayer 2004

kHz QPO & spin relation

Radio Pulsar: Magnetic field--period diagram PSR: 1750, X-ray NS: 200 magnetar: 5SGR+11AXP MSP: 175 BPSR: 130, recycled (2) 716 Hz; ~10^8 G; why not 10^7 G ? (1) Why B-P ? B evolves ? Recycled ? LMXB Van den Heuvel 2004, Science

Low frequency QPO---kHz QPO 关系 Belloni et al 2002 ApJ Empirical Relation ν HBO = 50. (Hz)(ν 2 /1000Hz) ν HBO = 42. (Hz) (ν 1 /500Hz) ν qpo = 10. (Hz) (ν 1 /500Hz) Low frequency QPO< 100 Hz FBO/NBO = 6-20 (Hz) HBO = (Hz) ν 1 = 700. (Hz)(ν 2 /1000Hz)

Low-high frequency QPO relation in WD/NS/BH Warner 2006 MNARS; Warner & Woudt 2004 MNRAS; Mauche 2002 ApJ; Titarchuk & Wood 2002 ApJ + 27 CVs, 5 magnitude orders in QPO frequency Black holes White dwarfs, Cvs Neutron stars ？ Zhang et al 2007, PASP Similarity in WD/NS/BH ?

Black Hole High Frequency QPOs HFQPO: (Hz) Frequency stable with Luminosity Pair relation 3:2 Frequency-Mass relation: 1/M Jets like Galactic BHs ~10Msun Different: NS/LMXB kHz QPOs ( McClintock & Remillard 2003;2006 ) Frequency at ISCO Schwarzschild ν k = (1/2π)(GM/r 3 ) 1/2 = (c/2πr) (R s /2r) 1/2 ν k ( ISCO ) = 2.2 (kHz) (M/Mסּ) -1 GRO J , XTE J XTE , 4U XTE , H GRS , 7 microquasars Van der Klis :2 mechanism: Abramowicz et al 2003; Li & Narayan 2003; Wang et al, 2003/06, MNRAS Measuring BH spin by QPO: Cui et al. 1998; Zhang SN et al 1997

High frequency pair QPO BH: ~3:2 --- NS: varied near 3:2

Theoretical Models Beat Model for KHz QPO ν 2 = ν kepler ν 1 = ν kepler - ν spin ∆ν = ν 2 - ν 1 = ν spin Miller, Lamb, Psaltis 1998; Strohmayer et al 1996 Lamb & Miller 2003 …Constant

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

Theoretical model Stella and Vietrie, 1999, Precession model Problems: 1.Vacuum 2.Circular orbit 3.Test particle 4.Predicted 2 M ⊙ sources, NS mass ~ 1.4 solar mass

Alfven wave oscillation MODEL Zhang 2004 AA; Li & Zhang 2005 ApJ; Keplerian Orbital frequency MHD Alfven wave Oscillation in the orbit ν 2 = 1850 (Hz) A X 3/2 ν 1 = ν 2 X (1- (1-X) 1/2 ) 1/2 A=m 1/2 /R 6 3/2; X=R/r, m: Ns mass in solar mass R 6 is NS radius in 10^6 cm Some radius: Lai 1998

Migliari, van der Klis, Fender, 2003 MNRAS Boutloukos ， van der Klis 2006 ApJ Cir X-1

Constrains star mass radius 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

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 NS 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, Kojim, Li XD, Xu RX, Zhang B, Kiziltan B 2007 MNRAS AqlX-1 ， EXO Samples

Fastest Pulsar XTE J Hz Mass & Radius Kaaret et al Quark Star ? Xu, MNRAS, 2005 Xu, Qiao, Wang, CPL, 2003 Li et al 1999 PRL Sub-millisecond PSR : high mass, Burderi et al. 2003

Estimating Magnetic filed - LMXB Spin > corotation radius ~40 km > ~10 8 G Spin variation: torque > ~10 8 G Wijnands & van der Klis 1998; Burderi, de Salvo ; Chou Y, poster kHz QPO distribution for Atoll/Z > similar magnetosphere ~10 8 G (Mdot) 1/2 Zhang & Kojim 2006; Burderi et al 1996, 1997; White & W Zhang 1998 Z source has a stronger field than Atoll’s

Estimating NS spin by spin-kHz QPO relation Spin frequency is less than the minimum upper-frequency of twin kHz QPOs

Summary 1. kHz QPOs NS/LMXBs 2. BH/WD/NS 3. Constraints M,R,B,spin