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The Neutron Star Equation of StateElectromagnetic Observations
Frits Paerels
Columbia University
GWPAW, UW Milwaukee, January 26, 2011
Planets: an Analogy
Measurements of mass and radius
Model M-R relation,
based on Equation of State
Courtesy Dimitar Sasselov/Harvard
Nature, 2008
Phase Diagram of H2O
Courtesy Dimitar Sasselov
Neutron Star Masses
single/double-lined binaries
+ optical vrad spectroscopy
of distorted star
Black Widow:
MPSR = 2.40 ± 0.12 MO
single/double-lined binaries
+ relativistic effects
single-lined binaries
+ relativistic effects
MNS = 1.97 ± 0.04 MO
Diagram from Lattimer&Prakash ‘What a Two Solar Mass Neutron Star Really Means’, 1012.3208v1
PSR + WD, i = 89°.17 (!!)
spectacular Shapiro delay:
clean mass measurement
(*) WD is point mass
DeMorest, Pennucci, Ransom, Roberts, Hessels, Nature, 467, 1083 (2010)
Neutron Star Radii
Neutron Star Radius
radio: X (radio emission not associated with NS surface)
If ~ blackbody:
optical: T = 5800 K, R = 10 km: MV 24 mag fainter than Sun;
at 100 pc: mV = 4.82 + 24 + 5 = 33.8 …
T = 106 K: gain 22 magnitudes; a few NS can be seen
If hotter, will be an X-ray source:
X-ray: Emax ~ 250 eV (T/106 K); L ~ LEdd for T ~ 107 K (for 1MO)
RX J1856.5-3754: indeed, sort of like a blackbody (kT ~ 60 eV)
Chandra LETGS: Drake et al., Ap.J., 572, 996 (2002)
Measuring the Mass and the Radius
1. Absolute Photometry: fν/Fν = (R/D)2 ; need Fν(Teff, log g, composition, B, …)
Need the distance D!
Also need to know what fraction of the stellar surface radiates!
The Magnificent Seven: seven soft X-ray sources with a ‘stellar’ spectrum and a
distance estimate
from Kaplan: 0801.1143
The interpretation of the photospheric spectrum is non-trivial:
from Kaplan: 0801.1143
Other attractive idea: use neutron stars in Globular Clusters (known D)
2. X-ray Burst Sources: go up to LEdd for 10 seconds, at T ~ 107 K
3. Periodically variable (spinning) X-ray bursters (‘hot spot’):
combine spin period, Doppler shift;
plus GR effects (lensing) on pulse shape: mass AND radius!
Currently, constrains (1 – RS/R)1/2 ; in future, M and R.
Burst oscillations in EXO0748-676
Galloway et al., Ap.J.(Letters), 711, L148 (2010)
Photospheric emission easily detectable.
If D known: same as previous.
4. Photospheric Spectroscopy
Most sensitive way to measure parameters: absorption line spectroscopy
a replay of classical stellar spectroscopy, with strong twists!
Ongoing accretion ensures ~ solar abundances
Expect: metals highly ionized, so focus on Fe
Line profiles sensitive to Doppler broadening, lensing, Lense-Thirring, …
Spin frequency 400 Hz
Full stellar surface
R/M = 4.82 G/c2
Özel and Psaltis, Ap.J.(Letters), 582, L31 (2003)
Line Profiles and Equivalent Widths
Doppler broadening of Fe: v/c = (kT/Mc2)1/2 = 1.3 x 10-4 (T/107 K)1/2
Absorption lines saturate, very hard to detect unless
spectroscopic resolving power > 5000
(NB. Stellar rotation does not affect [increase] the line contrast)
Easy to show that Stark broadening should easily be detectable:
ΔE ~ pE ~ (a0 e/Z) (e/r2) ~ n2/3 ~ g2/3
which is sensitive to density, hence to gravity!
Combine gravitational redshift with g, get M and R.
In practice, bursters spin rapidly, so cannot be done with current instruments
Baryonic EoS
So how far along are we?
M/R constraint
from X-ray
From Demorest et al., 2010
Free Quarks
‘Exotic’ condensates
Other techniques:
Precession of NS spin axis in binary: constrains moment of inertia I
PSR 0737-3039 A+B: binary pulsar, known masses;
geodetic precession of S around L with 71/75-yr period;
LS coupling introduces additional periastron advance
10% accuracy on I
Lattimer&Prakash: Phys.Reports, 2007
Prospects: spin-phase resolved photospheric spectroscopy with
the International X-ray Observatory IXO
And this will be multiply-redundant in M and R (also get redshift and g !)
XMM/RGS: Cumulative spectrum of 30 X-ray bursts
If correct identification: gravitational redshift!
z = 0.35
(Cottam, Paerels, & Mendez, 2002, Nature, 420, 51)