1. Constraining Neutron Star Radii and Equations of State Josh Grindlay Harvard (collaboration with Slavko Bogdanov McGill Univ.)

2. 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)

3. 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 ~ R 2 T 4 and measured T at “touchdown” when emission from (entire) NS surface Best done with LMXB in globular cluster, at well measured distance

4. 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

5. 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

6. 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 R NS = 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

7. ~50 MSPs detected in X-rays to date (mostly in globular clusters) Very faint X-ray sources - L X  10 33 ergs s –1 (0.1-10 keV) - typical: L X  10 30–31 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 Tuc Chandra ACIS-S 0.3-6 keV

8. e+e+ 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,…)

9. Modeling thermal X-ray emission from MSPs  Ingredients: - rotating neutron star - two X-ray  emitting 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)

10. Bogdanov, Grindlay, & Rybicki (2008) Synthetic MSP X-ray pulse profiles - R = 10 km, M = 1.4 M  - T eff = 2  10 6 K (H atmosphere) - 2 antipodal, point-like polar caps

11. Nollert et al. (1989) Flat Schwarzschild 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.

12. Bogdanov et al. (2007, 2008) 9 km 12 km 16 km for M = 1.4 M  * Fits to X-ray pulse profiles of MSPs can be used to infer NS compactness 1 + z g = (1 – 2GM/c 2 R) –1/2 (Pavlov & Zavlin 1997; Zavlin & Pavlov 1998) * Independent mass measurement for binary MSPs (e.g. PSR J0437  4715, 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

13. Neutron Star Hydrogen Atmosphere Model Courtesy of G.B. Rybicki BB H atm. Unmagnetized (B  10 8 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  =10  3

14. - P = 4 ms, R = 10 km, M = 1.4 M  - T eff = 2  10 6 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)

15. PSR J0437–4715 (nearest and brightest MSP) P = 5.757451924362137(99) ms D = 156.3  1.3 pc L X = 3  10 30 ergs s –1 M = 1.76  0.2 M  N H = 2  10 19 cm –2 Bogdanov, Rybicki, & Grindlay (2007) XMM–Newton EPIC-pn fast timing mode 0.3–2 keV 69 ks Black body H-atmos

16. Two-temperature H atmosphere T 1  2 × 10 6 K T 2  0.5 × 10 6 K R 1  300 m R 2  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 M  Bogdanov, Rybicki, & Grindlay (2007) 69 ks PSR J0437–4715

17. 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 T 1  1.5 × 10 6 K T 2  0.7 × 10 6 K R 1  400 m R 2  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

18. 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)

19. Modeling Thermal X-ray Emission from MSPs Most (?) Promising method for constraints on NS EOS: Extraordinary rotational stability (P =5.757451924362137(99) for J0437  4715) Non-transient (always “on”) and non-variable “Weak” magnetic fields (B surf ~10 8–9 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 (R eff  2 km)  emission region size and shape only important at  1% level High precision distances (  0.8% for PSR J0437  4715; Deller et al. 2008)  uncertainty in (R eff /D) 2 greatly reduced Independent, accurate mass measurements possible from radio timing  unique constraint on R

20. 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 !