Mass

1
Mass Radius of Compact Objects Fastest pulsar
and its stellar EOS

• CHENGMIN ZHANG
• National Astronomical Observatories
• Chinese Academy of Sciences, Beijing

2
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

3
Neutron Stars ?

• 40 NSs, M1.4 M? , R 10 -30 km ?
• Radio pulsars, X-ray NS, binary systems

(MT77)
(Stairs 2004)
(Lattimer Prakash 2004,2006)
4
NS mass determined in Binary system
MSP, PSR J07511807, M 2.1(2) M? ? Nice et
al. 2004 2A1822-371, Mgt0.97-0.24 M? Jonker et
al 2003 (1.74 M? ,2008) DNS M1.25M?, M1.34
M? , double pulsars (2004)
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PSR J0737-3039A/B Post-Keplerian Effects
R Mass ratio w periastron advance g
gravitational redshift r s Shapiro delay 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)
6
Measured M-R relations
No direct measure of radius !

• Apparent Radius R8R/(1-Rs/R)1/2
• Gravitational redshift z(1-Rs/R)-1/2 -1
• Mass density M/R3
• gM/R2
• 1E1207.4-5209, Aql X-1 and EXO 0748-676

Rs2GM Schwarzschild radius
7
Photon Spectra Key to Measuring Radius
For perfect Black Body Observed Total Flux F
4? R82 ?SB T84/d2
RX J1856.5-3754 (Fred Walters Star !)
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 ?
8
The Mass-Radius
Exotic Stars
Gravitational Red-shift observation of spectral
lines (Cottam, et al 2002).
QPOs indicate ISCO
9
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 ?2 1000
(Hz) Lower ?1 700 (Hz) Twin 21/27 sources
290
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Constrain star M_R by kHz QPOs

• Inner boundary to emit kHz QPO ISCO, R gt MAX M,
  R
• Mlt2.2 M? (1kHz/freq)
• Rlt19.5 km (1kHz/freq)
• M/R3 relation known by model for twin kHz QPOs
• SAXJ 1808.4 M/R3 by Burderi King 1998

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kHz QPOs from LMXBs R-ISCO
Sco X-1
Excluded
kHz QPO maximum frequency constrains NS equations
of state
12
Striking case of RX J1856.5-3754Truempet 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 lt10
km/M? Ransom et al 2004
? Apparent radius R816.5 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 )
13
Relativistic precession model by Stella Vietri
1999
M inferred from twin kHz QPOs Max frequency
ISCO
ISCO Saturation
Einsteins General Relativity Perihelion
precession Precession Model for KHz QPO,
Stella and Vietri, 1999 ?2 ?kepler ?1
?precession ?2 1 (1 3Rs/r)1/2 ??
?2 - ?1 is not constant
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M/R3 inferred from twin kHz QPOs Max frequency
Star Surface R
Kepler frequency ?k (GM/4p2r3)0.5 ?k 1850
(Hz) A X3/2 ?1 ? 2X (1- (1-X)1/2)1/2
A2m/R63 XR/r, mM/M? , R6 R/106 cm Zhang
2004, AA Li Zhang 2005
Maximum kHz QPO occurs at R or ISCO3Rs Agt ?k
/1850 (Hz) and m lt 2200 (Hz)/ ?k Miller et al
1998
15
Constraining M R by R8 and z

• 1E 1207.4-5209
• R84.6 km, Bignami et al 2004
• z0.12-0.23 Sanwal et al 2002 ?
• R 6 R86 /(1z)
• Mf(z)R86 /(1z)
• F(z)(20/3)z(1z/2)/(1z)2

16
Constraining M R by R8 and AM/R3

• Aql X-1
• 9 kmltR8lt18 km, Rutledge et al 2001
• one kHz QPO 1040 Hz van der Klis 2006
• R6 R86 /(10.15(A/0.7)2 R286 )0.5
• mAR36

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Constraining M R by AM/R3 and z

• EXo 0728-676
• z0.35 Cottam et al 2002
• One kHz QPO 695 Hz Homan van der Klis 2000
• R6 1.43f0.5(z)(0.7/A)
• m1.43f1.5(z)(0.7/A)
• f(z)(20/3)z(1z/2)/(1z)2

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1E1207.4-5209, Apparent radius, gravitational
redshift
QUARK STAR ?
19
Aql X-1 , Apparent radius14 km, single kHz QPO
20
EXO 0748-676 , gravitational redshift, kHz QPO
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Mass-Radius relations
Measuring NS Mass Radius by kHz QPO,
gravitational redshift and apparent radius

• Apparent Radius R8R/(1-Rs/R)1/2
• Haensel 2001
• Gravitational redshift z(1-Rs/R)-1/2 -1
• Cottam et al 2003, z0.35
• Mass density M/R3 (by kHz QPOs)
• Zhang 2004
• 1E1207.4-5209, Aql X-1 and EXO 0748-676

Rs2GM Schwarzschild radius
22
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
23
How about the Sub-millisecond Pulsar XTE
J1739285, spin1122 Hz

• Spin1122 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
24

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

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Max kHz QPO 1330 Hz
Zhang et al. 2006
Cir X-1
difference
Ratio
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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 MSPMax
Spin716 Hz
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kHz QPO spin relation
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• List of the Low-Mass X-Ray Binaries
  Simultaneously
• Detected Twin Kilohertz QPO and Spin
  Frequencies
• QPO (Hz)
  spin Dnu/spin
• 4U 160852 . . . . . . . . 8021099 619
  1.3
• 4U 163653 . . . . . . . . 9711192 581
  1.7
• 4U 170243 . . . . . . . . . 1055 330
  3.2
• 4U 172834 . . . . . . . . . 5821183 363
  1.6
• KS 1731260 . . . . . . . . 1169 524
  2.2
• 4U 191505 . . . . . . . . . 5141055 270
  1.9
• XTE J1807294 . . . . . . 353587 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 ?
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Summary

• Conclusions M-R relations
• Mass, measured
• Radius, not measured directly
• Spectra, MR relation
• Redshift, M/R
• kHz QPO, M/R3, constraints
• Others Ozel 2006

Not clear fuzzy in M-R EOS Quark or Neutron ?
THANKS
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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