Neutron Star Mass and Radius Measurements and

the Equation of State of Cold Dense Matter

Scott RansomNRAO / Univ of Virginia

Charlottesville,VA USA

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

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

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

NS Mass Measurements

From Lattimer & Prakash 2007

X-rayBinaries

DoubleNSs

MSPs

1.4Msun

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

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

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

Timing SensitivityTiming precision depends on:

– Sensitivity (A/Tsys)

– Pulse width (w)

– Pulsar flux density (S)

– Instrumentation

From Jenet & Demorest

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

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

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

M28C

Highly eccentric (e=0.847)Ppsr ~ 4.158 ms

Porb = 8.07 days

~5μs timing in ~5min

Mtot

=

1.631(2)M⊙

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⊙

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

Ter5I: Time dilation + redshift (gamma)

Highly eccentric (e=0.43)Ppsr ~ 9.57 ms

Porb = 31.87 hrs

Previouslyunmeasurable...

Convergingnicely...

Shapiro Delay

NRAO / Bill Saxton

Irwin Shapiro 1964Shapiro et al. 1968, 1971

Shapiro Delay with a MSP

NRAO / Bill Saxton

MSP B1855+09

Ryba & Taylor 1991 Kaspi, Ryba & Taylor 1994

Shapiro Delay

Amplitude ∝ Companion MassNarrowness ∝ Inclination

Conjunction

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

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°

Champion et al. 2008, Science, 320, 1309

~2 μs RMS

J1903+0327 (Eccentric MSP) Timing

Periastron precession +Shapiro delay

Recent Arecibotiming

Much improvedShapiro delay

PSR = 1.67(2) M⊙

Freire et al. 2011,MNRAS, 412, 2763

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

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

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

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!

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

MSP J1614-2230: EOS Constraints

Demorest et al. 2010, Nature, 467, 1081D see Ozel et al. 2010, ApJL, 724, 1990

NGC6544B

Highly eccentric (e=0.747)Ppsr ~ 4.18 ms

Porb = 9.96 daysLynch et al. in prep

Mtot

=

2.56(1)M⊙

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

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

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

Currently 36 new Radio/gamma-ray MSPs because of Fermi!

Courtesy: Paul Ray

~10% of them look like they will be “good timers”

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!)