Constraining Neutron Star Radii

and Equations of State

Josh GrindlayHarvard

(collaboration with Slavko Bogdanov McGill Univ.)

Outline of talk

Radii from X-ray bursts (BB fits)

Radii from quiescent LMXBs (BB fits)

Radii of isolated NSs (e.g. RXJ1856-3754)(J. Truemper’s talk…)

Radii from MSPs (M/R from light bending)

NS Radii from X-ray bursts

Type I x-ray bursts are thermonuclear flashes on NSs in low mass X-ray binaries (LMXBs)

Some are Eddington limited (flat-topped Lx) with BB radii determined from Lx ~ R2 T4 and measured T at “touchdown” when emission from (entire) NS surface

Best done with LMXB in globular cluster, at well measured distance

Radius Expansion X-ray burst from M15

M15 burst seen from X2127+119 by RXTE from M15 (d = 10 ±0.5 kpc) by Smale (2001):

Derived NS parameters: R* = 8.6 ±1km (but uncertain by Comptonizing atmosphere model) 1 + z = 1.28 ±0.06 and mass of NS = 2.38 ±0.18 Msun

vs. Spectral line shifts in X-ray burst

Cottam et al (2002, Nature) observed and stacked 28 bursts from EXO 0748-676

Candidate Fe XXVI lines seen at redshift z = 0.35

Atmospheric radii of quiescent LMXBs

Heinke et al (2006, ApJ) derive constraints on luminous quiescent LMXB X7 in 47Tuc, using NS-atmosphere model of Rybicki et al

Derived RNS = 14.5 ±1.7 km

for M = 1.4Msun

1 + z = 1.26 ±0.12

or if R = 10 km M = 2.20 ±0.1Msun

• ~50 MSPs detected in X-rays to date (mostly in globular clusters)

• Very faint X-ray sources - LX

1033 ergs s–1 (0.1-10 keV) - typical: LX 1030–31 ergs s–1

• Many exhibit (pulsed) soft, thermal X-ray emission from magnetic polar caps

Rotation-powered (“recycled”) millisecond pulsars

Bogdanov et al. (2006)

MSPs are “ideal”: Constant, noise free Binary companions

(allow mass meas.)

R

Y

19 MSPs in 47 TucChandra ACIS-S

0.3-6 keV

e+

e+

X-rays

Thermal X-ray emission due to polar cap heating by a return current of relativistic particles from pulsar magnetosphere

X-rays

The surface radiation can serve as a valuable probe of neutron star properties (compactness, magnetic field geometry, surface composition,…)

Modeling thermal X-ray emission from MSPs

Ingredients: - rotating neutron star

- two X-rayemitting hot spots

- General & special relativity * Schwarzschild metric

(good for 300 Hz)* Doppler boosting/aberration

* propagation time delays

- optically-thick hydrogen atmosphere

Viironen & Poutanen (2004)

= pulsar obliquity

= b/w line of sight & pulsar spin axis

(t) = rotational phase

= photon w.r.t surface normal

= photon at infinity

b = photon impact parameter at infinity

Viironen & Poutanen (2004)

Bogdanov, Grindlay, & Rybicki (2008)

Synthetic MSP X-ray pulse profiles - R = 10 km, M = 1.4 M

- Teff = 2 106 K (H atmosphere)

- 2 antipodal, point-like polar caps

Nollert et al. (1989)

FlatSchwarzschild

Gravitational redshift & bending of photon trajectories

For M = 1.4 M, R = 10 km ~80% of the entire neutron star surface is visible at a given instant.

Bogdanov et al. (2007, 2008)

9 km12 km16 km

for M = 1.4 M

* Fits to X-ray pulse profiles of MSPs can be used to infer NS compactness

1 + zg = (1 – 2GM/c2R)–1/2 (Pavlov & Zavlin 1997;Zavlin & Pavlov 1998)

* Independent mass measurement for binary MSPs (e.g. PSR J04374715, M=1.76 0.2 M)

constrain R separately

tight constraint on NS EOS

}=10°, =30°

=30°, =60°

=60°, =80°

=20°, =80°

Model MSP X-ray pulse profiles: Constraints on the NS EOS

Neutron Star Hydrogen Atmosphere Model

Courtesy of G.B. Rybicki

BB

H atm.

• Unmagnetized (B108 G ~ 0), Optically-Thick Hydrogen Atmosphere:

- 100% pure hydrogen due to gravitational sedimentation

- harder than blackbody for same effective temperature

- energy-dependent limb darkening

}Zavlin et al. (1996)

cos=0

cos=103

- P = 4 ms, R = 10 km, M = 1.4 M

- Teff = 2 106 K (H atmosphere)

- 2 antipodal, point-like polar caps

Blackbody

Blackbody + Doppler

H atmosphere

H atmospere + Doppler

Due to limb-darkening,

H atmosphere pulse profiles

differ substantially from

Blackbody and are required

=10°, =30°

=30°, =60°

=60°, =80°

=20°, =80°

Model MSP X-ray pulse profiles: H atmosphere vs blackbody

(see Pavlov & Zavlin 1997;

Zavlin & Pavlov 1998;

Bogdanov et al. 2007, 2008)Bogdanov et al. (2007)

PSR J0437–4715 (nearest and brightest MSP)

P = 5.757451924362137(99) ms D = 156.3 1.3 pc LX = 3 1030 ergs s–1

M = 1.76 0.2 M NH = 2 1019 cm–2

Bogdanov, Rybicki, & Grindlay (2007)

XMM–Newton EPIC-pnfast timing mode

0.3–2 keV69 ks Black body

H-atmos

Two-temperature H atmosphere

T1 2 × 106 K T2 0.5 × 106 K

R1 300 m R2 2 km

Inconsistent with blackbody

H atmosphere + centered dipole

Offset dipole required (~1 km)

R = 8.5–17.6 km (95% confidence)

R measured since

R > 8.5 km (99.9% confidence)

for M = 1.76 MBogdanov, Rybicki, & Grindlay (2007)

69 ks

PSR J0437–4715

Bogdanov & Grindlay in prep.

PSR J0030+0451

R > 10.6 km (95% conf.)

R > 10.4 km (99.9% conf.)

Lower limits since angles α, ζ not fixed

for M = 1.4 M

Two-temperature H atmosphere

T1 1.5 × 106 K T2 0.7 × 106 K

R1 400 m R2 1.5 km

Inconsistent with blackbody

H atmosphere required

Evidence for offset dipole

Nearby (D 300 pc) isolated MSP

XMM–Newton EPIC pn

130 ks

Constraints on M/R for MSP J0030+0451

95% conf. limits:

For M ≥1.4Msun

R ≥ 10.6km

Rules out Quark

Star models

SQM1, SQM3

(Bogdanov &

Grindlay 2009)

Modeling Thermal X-ray Emission from MSPs

• Most (?) Promising method for constraints on NS EOS:

Extraordinary rotational stability (P =5.757451924362137(99) for J04374715)

Non-transient (always “on”) and non-variable

“Weak” magnetic fields (Bsurf~108–9 G) B-field does not affect radiative properties of atmosphere

Dominant thermal emission (95% of total counts @ 0.1–2 keV)

Radiation from small fraction of NS surface(Reff 2 km) emission region size and shape only important at 1%

level

High precision distances (0.8% for PSR J04374715; Deller et al. 2008) uncertainty in (Reff/D)2 greatly reduced

Independent, accurate mass measurements possible from radio timing unique constraint on R

Conclusions

Bursts involve time-variable phenomena; not ideal but provide interesting constraints on M/R

qLMXBs in “purely thermal” state (without complications of hard-emission components found from PWN and/or propeller effect contributions) give more reliable M/R

MSPs with thermal polar cap emission offer best M/R constraints

MSP J0437-4715 is a clean (WD-NS) binary. Shapiro delay timing will give M; angles α, ζ can be measured. Actual values of M and R can/will be obtained !