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

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 ？
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40+ NSs, M=1.4 M⊙ , R= 10 -30 km ?
Radio pulsars, X-ray NS, binary systems
(MT77)
(Stairs 2004)
(Lattimer & Prakash 2004，2006)
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)
PSR J0737-3039A/B Post-Keplerian Effects
R: Mass ratio
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w: periastron advance
g: gravitational redshift
r & s: Shapiro delay
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Pb: orbit decay
• Six measured
parameters – only two
independent
• Fully consistent with
general relativity (0.1%)
A: 1.34 M⊙ ; B: 1.25 M⊙
(Kramer et al. 2005)
No direct measure of radius !
Measured M-R relations
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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 0748-676
Rs=2GM: Schwarzschild radius
Photon Spectra: Key to Measuring Radius
For perfect Black Body:
Observed Total Flux: F =4 R∞2 SB T∞4/d2
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 !)
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R 
R
1- 2GM/Rc2
T  1- 2GM/Rc2 T
( 1- 2GM/Rc 2  (1 z) -1 )
The Mass-Radius
Exotic Stars
Gravitational Red-shift:
observation of spectral
lines (Cottam, et al 2002).
QPOs indicate ISCO
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
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Inner boundary to emit kHz QPO: ISCO, R > MAX M, R
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M<2.2 M⊙ (1kHz/freq)
R<19.5 km (1kHz/freq)
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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
Sco X-1
Excluded
kHz QPO maximum frequency constrains
NS equations of state
Striking case of RX J1856.5-3754
Truempet et al. 2004; Burwitz et al. 2003
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
 Apparent radius R∞=16.5 km (d/117pc), Truempet 2005
 True radius 14 km (1.4 M⊙), stiff EOS, rule out quark star
Relativistic precession model by Stella & Vietri 1999
M inferred from twin kHz QPOs
Max frequency – ISCO
ISCO Saturation
Einstein’s General Relativity: Perihelion precession
Precession Model for KHz QPO, Stella and Vietri, 1999
ν2 = νkepler
ν1 = νprecession
∆ν = ν2
= ν2 [1 – (1 – 3Rs/r)1/2]
- ν1 is not constant
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
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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
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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
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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
Measuring NS Mass & Radius
Mass-Radius relations
by kHz QPO, gravitational redshift and apparent radius
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Apparent Radius: R∞=R/(1-Rs/R)1/2
Haensel 2001
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Gravitational redshift: z=(1-Rs/R)-1/2 -1
Cottam et al 2003, z=0.35
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Mass density: M/R3 (by kHz QPOs)
Zhang 2004
1E1207.4-5209, Aql X-1 and EXO 0748-676
Rs=2GM: Schwarzschild radius
Measuring STAR Mass-Radius
by kHz QPO, gravitational redshift and apparent radius
Zhang, Yin, Li, Xu, Zhang B, 2007
AqlX-1， EXO 0748-676 Samples
CN1/CN2: normal neutron matter, CS1/CS2: quark star
CPC: Bose-Einstein condensate of pions
How about the Sub-millisecond Pulsar
XTE J1739285, spin=1122 Hz
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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……
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ISCO condition, m ≤ 2200 (Hz)/spin
Keplerian at R, crust split
Max kHz QPO 1330 Hz
Ratio
difference
Zhang et al. 2006
Cir X-1
Spin Frequency - LMXBs
Spin frequency:
Max: 1122 Hz, Kaaret et al 2007
Min: 45 Hz Villarreal & Strohmayer 2004
23 Spin sources, Av ~ 400 Hz
Radio MSP：Max Spin=716 Hz
kHz QPO & spin relation
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List of the Low-Mass X-Ray Binaries Simultaneously
Detected Twin Kilohertz QPO and Spin Frequencies
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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)
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Fastest Pulsar XTE J1739-285
spin = 1122 Hz M – R Kaaret et al. 2007
Quark Star =
sub-MSP ?
Quark Star ?
Summary
Conclusions: M-R relations
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Mass, measured
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Radius, not measured directly
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Spectra, MR relation
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Redshift, M/R
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kHz QPO, M/R^3, constraints
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Others… Ozel 2006
THANKS
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