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Neutron Star Mass and Radius Measurements and
the Equation of State of Cold Dense Matter
Scott RansomNRAO / Univ of Virginia
Charlottesville,VA USA
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Why Measure NS Masses / Radii?• Most dense objects known in the Universe• Mass – radius relation is determined by EoS• Stress Nuclear Physics and Standard Model
From Lattimer & Prakash 2007
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X-rays can constrain both at once• Surface Black Body thermal emission (R and Temp)• Emission near surface shows light-bending and
gravitational redshift (can give M/R)
Ozel 2006, Nature, 441, 1115
• Thermonuclear X-ray bursts can achieve Eddington luminosity (i.e radiation pressure out equals gravitational pressure in)
• Measure multiple phenomena at once
• Problems are systematics and photon statistics
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Main X-ray techniques / sources• Thermonuclear Radius Expansion Bursts
• Eddington Limit + Surface Emission• Ozel et al 2009 ApJ / Guver et al 2010 ApJ
• Thermal Pulsations from X-ray Millisecond Pulsars• Light-bending, redshift, Doppler boosting / aberration (M/R)
• Pavlov & Zavlin 1997 / Zavlin & Pavlov 1998 / Bogdanov et al 2007 ApJ
BB
Hatm
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NS Mass Measurements
From Lattimer & Prakash 2007
X-rayBinaries
DoubleNSs
MSPs
1.4Msun
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Non-recycled
Recycled: Binaries Isolated Double-NSs
Pulsar Population
~2000 pulsars known~140 in Glob. Clusters
In Galaxy: ~90 binary MSPs ~30 isolated MSPs
~25 binary part-recyc ~15 isolated part-recyc
Definitions: Part-recycled: P > 20 ms, B < 3x1010 G MSP: P < 20 ms, B < 109 G
Millisecond Pulsars (MSPs)
NormalPulsars
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Obs 2
Pulsar Timing:Unambiguously account for every
rotation of a pulsar over years
Observation 1
Pulses
Obs 3
Model(prediction)Measurement
(TOAs: Times of Arrival)
Measurement - Model = Timing Residuals
200ns RMSover 2 yrs
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Precision Timing Example
• Astrometric Params• RA, DEC, μ, π
• Spin Params• Pspin, Pspin
• Keplerian Orbital Params• Porb, x, e, ω, T0
• Post-Keplerian Params• ω, γ, Porb, r , s
~100 ns RMStiming residuals!
van Straten et al., 2001 Nature, 412, 158Recent work (e.g. Verbiest et al 2009) shows
this is sustainable over 5+ yrs for several MSPs
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Timing SensitivityTiming precision depends on:
– Sensitivity (A/Tsys)
– Pulse width (w)
– Pulsar flux density (S)
– Instrumentation
From Jenet & Demorest
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Besides the normal 5 “Keplerian” parameters (Porb, e, asin(i)/c, T0, ω), General Relativity gives:
where: T⊙ ≡ GM⊙/c3 = 4.925490947 μs, M = m1 + m2, and s ≡ sin(i)
Post-Keplerian Orbital Parameters
(Orbital Precession)
(Grav redshift + time dilation)
(Shapiro delay: “range” and “shape”)
These are only functions of:- the (precisely!) known Keplerian orbital parameters P
b, e, asin(i)
- the mass of the pulsar m1 and the mass of the companion m
2
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Besides the normal 5 “Keplerian” parameters (Porb, e, asin(i)/c, T0, ω), General Relativity gives:
where: T⊙ ≡ GM⊙/c3 = 4.925490947 μs, M = m1 + m2, and s ≡ sin(i)
Post-Keplerian Orbital Parameters
(Orbital Precession)
(Grav redshift + time dilation)
(Shapiro delay: “range” and “shape”)
These are only functions of:- the (precisely!) known Keplerian orbital parameters P
b, e, asin(i)
- the mass of the pulsar m1 and the mass of the companion m
2
Need eccentric orbit andtime for precession
Need compact orbit and a lot of patience
Need high precision,Inclination, and m
2
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Precession of Periastron• Gives total system mass
• Mercury is 42”/century
• DNS systems are deg/yr
• Need eccentric system
• “Easy” to measure
• If orbits are random, distribution is flat in cos(i)
• Possible (unlikely?) classical contributions (i.e. rotating WD, tidal effects) From new MSP Terzan5ai
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M28C
Highly eccentric (e=0.847)Ppsr ~ 4.158 ms
Porb = 8.07 days
~5μs timing in ~5min
Mtot
=
1.631(2)M⊙
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NGC6440B: A Massive PSR?
Highly eccentric (e=0.570)Ppsr ~ 16.76 ms
Porb = 20.6 days
Mtot
=
2.8(3)M⊙
Freire et al. 2008, ApJ, 675, 670
Mc ~
0.1 M⊙
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13 Eccentric (e>0.3) PSRs in ClustersName P(ms) Pb(d) E
Ter5ai 21.228 0.85 0.440 0.49 1.883(4) 1.39
Ter5J 80.338 1.10 0.350 0.34 2.19(2) 1.73
Ter5I 9.570 1.33 0.428 0.21 2.171(3) 1.87
Ter5Z 2.463 3.49 0.761 0.22 1.79(1) 1.53
Ter5U 3.289 3.57 0.605 0.39 2.26(1) 1.73
Ter5X 2.999 5.00 0.302 0.25 1.91(5) 1.60
M5B 7.947 6.85 0.138 0.13 2.3(1) 2.12
M28C 4.158 8.08 0.847 0.26 1.631(1) 1.33
NGC6441A 111.601 17.33 0.712 0.59 2.0(2) 1.35
NGC1851A 4.991 18.79 0.888 0.92 2.44(5) 1.34
NGC6440B 16.760 20.55 0.570 0.08 2.8(3) 2.68
Ter5Q 2.812 30.30 0.722 0.46 2.4(2) 1.79
M28D 79.835 30.41 0.776 0.38 1.2(7)
Mcmin Mtot Mpmed
M5B: Freire et al 2008, ApJ, 679, 1433
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Ter5I: Time dilation + redshift (gamma)
Highly eccentric (e=0.43)Ppsr ~ 9.57 ms
Porb = 31.87 hrs
Previouslyunmeasurable...
Convergingnicely...
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Shapiro Delay
NRAO / Bill Saxton
Irwin Shapiro 1964Shapiro et al. 1968, 1971
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Shapiro Delay with a MSP
NRAO / Bill Saxton
MSP B1855+09
Ryba & Taylor 1991 Kaspi, Ryba & Taylor 1994
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Shapiro Delay
Amplitude ∝ Companion MassNarrowness ∝ Inclination
Conjunction
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PSR J1903+0327 with Arecibo P-ALFA
This thing is weird.- Fully recycled PSR- Highly eccentric orbit- Massive main-sequence
star companion- Massive NS (1.67 Msun)
(Freire et al. 2011)
- High precision timing despite being distant and in Galactic plane
Bill Saxton, NRAO/AUI/NSF
Champion et al. 2008, Science, 320, 1309
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J1903+0327 (Eccentric MSP) Timing
One Orbit (Orbital Phase)
Champion et al. 2008, Science, 320, 1309
Pspin = 2.15 msPorb = 95 daysEccen = 0.436 (!)Mc > 0.905 Msunl,b = 37.3°, -1.0°
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Champion et al. 2008, Science, 320, 1309
~2 μs RMS
J1903+0327 (Eccentric MSP) Timing
Periastron precession +Shapiro delay
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Recent Arecibotiming
Much improvedShapiro delay
PSR = 1.67(2) M⊙
Freire et al. 2011,MNRAS, 412, 2763
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Two Mass Distributions?
Kiziltan, Kottas & Thorsett (1011.4291)See also
Zhang et al., 2011, A&A, 527, 83Valentim, Rangel & Horvath, 2011,
MNRAS, 414, 1427
• Bayesian Analyses suggest bi-modal mass distribution
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Empirically Determined EoSSteiner, Lattimer & Brown, 2010, ApJ, 722, 33
• Use all astrophysical evidence to date (esp. NS masses, roration, and X-ray Radius Expansion Bursts)
• Parameterized EoS via Bayesian analysis• NSs have ~12 km radii, and max mass is >2 Msun
M vs R
Pressure
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MSP J1614-2230• 3.15 ms in 8.7 day binary
• Gamma-ray pulsar
• >0.4 M⊙ companion
• Orbital timing systematics from past years?
“GUPPI”Coherent De-Dispersion
1100-1900 MHz
~10 us residuals
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GUPPI: Coherent Dedispersion● FPGAs feeds 8 gaming
systems w/ NVIDIA GPUs (programmed by P Demorest)
● Large improvement in timing precision (i.e. for NANOGrav)
● Clone for Arecibo and software version for EVLA this summer....
~1 TFLOP inreal-time!
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MSP J1614-2230:Incredible Shapiro
Delay Signal
Demorest et al. 2010, Nature, 467, 1081D see Ozel et al. 2010, ApJL, 724, 1990
Mwd
= 0.500(6) M⊙
Inclination = 89.17(2) deg!
Mpsr = 1.97(4) M⊙!
Full Shapiro Signal
No General Relativity
Full Relativistic Solution
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MSP J1614-2230: EOS Constraints
Demorest et al. 2010, Nature, 467, 1081D see Ozel et al. 2010, ApJL, 724, 1990
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NGC6544B
Highly eccentric (e=0.747)Ppsr ~ 4.18 ms
Porb = 9.96 daysLynch et al. in prep
Mtot
=
2.56(1)M⊙
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1.6 ms pulsar in 9 hr orbit
~20-30 MJup
companion
Make isolated MSPs?
Fruchter, Stinebring & Taylor, 1988, Nature, 333, 237
Since then:Orbital variabilityHigh velocity (>220km/s)Optical bowshockOptical modulationX-ray tail
Stappers et al. 2003, Science, 299, 1372
Fruchter et al. 1990, ApJ, 351, 642
Original “Black Widow”: B1957+21
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Original “Black Widow”: B1957+21• New radial vel curve:
353(4) km/s amplitude (corr. for ctr-of-light)
• i=65(2)deg from lightcurve models
• Mp ~ 2.40+/-0.12Msun
• Mp > 1.66 Msun
van Kerkwijk, Breton, & Kulkarni, 2011 ApJ, 728, 95
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Year
Cumulative Number of Known MSPs in Galactic Disk
Num
b er
of M
SP
sNumbers have:
quadrupled in last 10 yrsdoubled in last 4 years
Why?Rise in computing capability, sensitive new radio surveys, Fermi!
120
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Currently 36 new Radio/gamma-ray MSPs because of Fermi!
Courtesy: Paul Ray
~10% of them look like they will be “good timers”
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Summary!• X-ray observations are very promising for the future
• Need new timing capability in space, though!
• Timing observations of binary pulsars can allow precise neutron star mass measurements
• Two new ones are: 1.667(21) and 1.97(4) Msun
• No more “canonical” 1.4 Msun NS
• Require Post-Keplerian parameters
• Need very high-precision timing
• Eccentric orbits and massive companions help a lot
• We know of ~200 MSPs many more to find
• Only a few percent might provide good masses
• Current surveys completely sensitivity limited (Fermi helps!)