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= 10 -30 km ?

Radio pulsars, X-ray NS, binary systems

NS mass determined in Binary system

MSP, PSR J0751+1807, M = 2.1(2) M ⊙ ？； Nice et al. 20042A1822-371, 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)⊙

PSR J0737-3039A/B Post-Keplerian Effects

R: Mass ratio

: periastron advance

: gravitational redshift

r & s: Shapiro delay

Pb: 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-Rs/R)1/2

Gravitational redshift: z=(1-Rs/R)-1/2 -1 Mass density: M/R3 g=~M/R2

1E1207.4-5209, Aql X-1 and EXO 0748-676Rs=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/d2

R R

1- 2GM/Rc 2

T 1- 2GM/Rc 2 T

( 1- 2GM/Rc 2 (1 z)-1)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 !)

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 ν2 ~ 1000 (Hz)

Lower ν1 ~ 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/R3 relation known by model for twin kHz QPOs

SAXJ 1808.4: M/R3 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 J1856.5-3754

Truempet et al. 2004; Burwitz et al. 2003

Apparent radius RApparent radius R∞∞=16.5 km (d/117pc), =16.5 km (d/117pc), Truempet 2005Truempet 2005 True radius 14 km (1.4 MTrue radius 14 km (1.4 M⊙⊙), stiff EOS, rule out quark star), stiff EOS, rule out quark star

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

ν2 = νkepler

ν1 = νprecession = ν2 [1 – (1 – 3Rs/r)1/2]

∆ν = ν2 - ν1 is not constant

ISCO Saturation

Relativistic precession model by Stella & Vietri 1999

M inferred from twin kHz QPOs

Max frequency – ISCO

M/R3 inferred from twin kHz QPOs

Max frequency – Star Surface R

Kepler frequency νk = (GM/4π2r3)0.5

νk = 1850 (Hz) A X3/2

ν1 = ν 2X (1- (1-X)1/2)1/2

A2=m/R63; X=R/r, m=M/M⊙ , R6 = R/106 cm

Zhang 2004, AA; Li & Zhang 2005

Maximum kHz QPO occurs at R or ISCO=3Rs

A> νk /1850 (Hz) and m < 2200 (Hz)/ νk

Miller et al 1998

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

Constraining M – R by R∞ and A~M/R3

Aql X-1 : 9 km<R∞<18 km, Rutledge et al 2001

one kHz QPO: 1040 Hz; van der Klis 2006

R6 =R∞6 /(1+0.15(A/0.7)2 R2∞6 )0.5

m=AR36

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

R6 =1.43f0.5(z)(0.7/A) m=1.43f1.5(z)(0.7/A) f(z)=(20/3)z(1+z/2)/(1+z)2

1E1207.4-5209,

Apparent radius, gravitational redshift

QUARK STAR ?

Aql X-1 ,

Apparent radius=14 km, single kHz QPO

EXO 0748-676 ,

gravitational redshift, kHz QPO

Mass-Radius relations

Apparent Radius: R∞=R/(1-Rs/R)1/2 Haensel 2001

Gravitational redshift: z=(1-Rs/R)-1/2 -1 Cottam et al 2003, z=0.35

Mass density: M/R3 (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 　　

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

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……

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

Zhang et al. 2006

Max kHz QPO 1330 Hz

Cir X-1

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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 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)

Fastest Pulsar XTE J1739-285 spin = 1122 Hz M – R Kaaret et al. 2007

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

4U1820-30, NASA

Swank 2004; Miller 2004

BH/ISCO: 3 Schwarzschild radius

Innermost stable circular orbit

NS/Surface: star radius, hard surface