NICER Constraints on the Dense Matter Equation of State
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Measuring the neutron star equation of state with NICER. https://https://iopscience.iop.org/journal/2041-8205/page/Focus on NICER collaboration, Amsterdam and USA consortia. October 5, 2020 – Ecogia Science Meetings 2020–2021
Measuring the neutron star equation of state with NICER. Preamble Table of Contents Introduction NICER The source Methods Results Riley et al. (2019) Results from Miller et al. (2019) Implications for the equation of state Implication for NS physics October 5, 2020 – Ecogia Science Meetings 2020–2021 1
Measuring the neutron star equation of state with NICER.
Introduction
Neutron stars
• Outcome of supernova explosion
• Typical masses of 1–2M , radii of
10–15 km
• solid ionic crust supported by
electron degeneracy pressure.
• Neutrons begin to leak out of ions
(nuclei) at densities
∼ 4 × 1011 g/cm3 (the neutron
drip density, which separates the
inner from the outer crust), where
Watts et al. 2016, http: // link. aps. org/ doi/ 10. neutron degeneracy also starts to
1103/ RevModPhys. 88. 021001 play a role.
• At densities ∼ 2 × 1014 g/cm3 , the
nuclei dissolve completely. This
marks the crust-core boundary.
October 5, 2020 – Ecogia Science Meetings 2020–2021 2
Measuring the neutron star equation of state with NICER.
Introduction
Equation of state
• The equation of state links pressure and density of matter
• For a perfect gas p = kb ρT , for dense matter p = K ρΓ , with Γ ∼ 2
µmu
temperature effect negligible
• It is used in integrating the internal Tollman-Oppenhaimer-Volkoff
equation, the relativistic hydrostatic equilibrium equation
dp (p + r )(m + 4πr 4 p)
=
dr r 2 (1 − 2m/r )
• The radius depends on mass, for self gravitation.
• Note that rotation should be considered
October 5, 2020 – Ecogia Science Meetings 2020–2021 3
Measuring the neutron star equation of state with NICER.
Introduction
Methods to derive mass and radius
• Gravitational redshift of an absorption/emission line on the NS surface
(one controversial claim, Cottam et al. 2002)
−1/2
2GM
1+z = 1−
C 2R
• Gravitational waves: the final stages of tidal interactions are sensitive to
the equation of state (a poor constraint)
• Pulses from accretion-powered pulsars, but highly variable and with
unknown details on streaming of matter.
• Burst oscillations: ignition of thermal flames in the initial phase of
thermal bursts. Uncertainties in the place of the ignition.
October 5, 2020 – Ecogia Science Meetings 2020–2021 4
Measuring the neutron star equation of state with NICER. Introduction Pulse fitting • Light bending due to general relativistic effects modifies subtly the pulse profiles, even for point source on the surface • the visible surface is larger than in flat space-time • with very high signal to noise, it is possible to disentangle the effect of the compactness ratio (M/R) October 5, 2020 – Ecogia Science Meetings 2020–2021 5
Measuring the neutron star equation of state with NICER.
NICER
NICER
• The Neutron star Interior Composition Explorer (NICER) is mounted
on the ISS
• NICER/XTI consists of an array of 52 active silicon drift detectors
housed in focal plane modules(FPMs), each paired with a nested
single-reflection grazing-incidence “concentrator” optic assembly in the
optical path
• the XTI’s concentrator optics are co-aligned, collecting sky emission
from a single≈3’ radius non-imaging FOV.
• The instrument is sensitive to X-rays in the 0.2–12 keV band, with a
peak effective area of ≈ 1900 cm2 around 1.5 keV
• Core science to collect tens of Ms of data on pulsars
• timing accuracy ∼10 ns
• time stamp 90 ns (fast chain) or
256 ns (slow chain, E < 1 keV)
• response accuracy: 2%
October 5, 2020 – Ecogia Science Meetings 2020–2021 6
Measuring the neutron star equation of state with NICER. The source Rotation-powered pulsars • A rotation powered pulsar: pulsed emission from thermal radiation from the surface at T ∼ 106 K. • Radiation is produced by a backflow of energetic particles along the magnetic-field lines • a non-magnetic hydrogen atmosphere can reproduce the energy-dependent X-ray pulse profiles of the two closest known MSPs, PSRs J0437−4715 and J0030+0451 using XMM-Newton • large-amplitude pulsations are incompatible with a model that considers an isotropically emitting Planck spectrum October 5, 2020 – Ecogia Science Meetings 2020–2021 7
Measuring the neutron star equation of state with NICER. The source PSR J0030+0451 • a solitary MSP was discovered at radio frequencies in the Arecibo drift scan survey • spin period P=4.87 ms and intrinsic spindown rate Ṗ = 1.02 × 10−20 s s−1 • B ≈ 2.7 × 108 G, a characteristic age τ = 7.8 Gyr Ė = 3 × 1033 erg/s • X-ray spectrum compatible with a two-temperature thermal plus hard tail above 3 keV • two broad pulses with pulsed fraction 60–70% → beaming of emission October 5, 2020 – Ecogia Science Meetings 2020–2021 8
Measuring the neutron star equation of state with NICER. The source Data-set • NICER pointed at the source between 2017 July 24 and 2018 December 9 • filtered and phase-folded data in Zenodo • 1.936 Ms of exposure in the 0.25–1.45 keV range (which yields the highest pulsed signal detection significance of 172.8σ) • no energy-dependent phase shift • check the barycentric correction with two independent methods • long-term timing is assured by radio monitoring and pulsar stability and checked with phase folding in NICER • model nsatmos; no spectral variability • pulse profiles have four significant harmonic components October 5, 2020 – Ecogia Science Meetings 2020–2021 9
Measuring the neutron star equation of state with NICER.
Methods
The journey of photons
• For a rotating neutron star, it is necessary to consider relativistic effects:
use of a Schwartschild external space-time and oblate solution for the NS
surface.
1. consider the local Doppler boosting in passing from the local comoving
frame (the particles on the NS surface) to the local static frame
2. compute a look-up table to connect the deflection angle
ψ = cos−1 [(θ, φ) · (ζ, φobs )] to the angle α from the surface normal in the
local static frame so that a photon leaving the surface at that angle will be
deflected by an angle ψ in propagating to infinity (light deflection)
3. compute the time delays for different ψ
4. compute the observed flux as a product of a lensing factor and of Doppler
boosting
October 5, 2020 – Ecogia Science Meetings 2020–2021 10Measuring the neutron star equation of state with NICER. Methods Oblatness and codes • oblatness is introduced with a convenience formula R(θc ) = Req [1 + o2 (x, Ω̄) cos2 (θc )] • checks are done for accuracy using different equation of state and numerical solution for accuracy < 0.1%. • Adapted equation because the radial direction is different from the surface normal • compute the surface gravity relevant for emission mechanisms • At least 6 different codes were checked for consistency and the deviations for the oblate plus grid from the best numerical simulation is less than 0.1% (Note that using non-oblate stars introduce an error of 1.5%) October 5, 2020 – Ecogia Science Meetings 2020–2021 11
Measuring the neutron star equation of state with NICER. Methods Modeling details • Two separated groups made independent analysis: Riley et al. (2019) and Miller et al. (2019). • They both use a Bayesian approach to model the energy-dependent pulse profiles as function of several parameters, including mass and radius of the NS • geometrically thin fully ionized hydrogen atmosphere which characterizes the thermal emission of hot regions • Gaussian prior for the distance based on radio observations (only Riley et al. 2019). • a joint prior distribution of mass and radius that facilitates the subsequent inference of EOS model parameters • uncertainty on response both in absolute calibration and energy-dependent effective are, parametrized based on Crab observations (only Riley et al. 2019) • differences between groups are in the hot region configurations, the instrumental response, and the specification of the prior on distance. • pulse is double peaked, so they use at least two hot “spots” with increasing complexity for the shape and temperature function of the hot regions October 5, 2020 – Ecogia Science Meetings 2020–2021 12
Measuring the neutron star equation of state with NICER. Methods Handling complexity They use a combination of performance measures: • the evidence (the prior predictive probability of the data) • graphical posterior predictive checking (to verify whether or not a model generates synthetic data without obvious residual systematic structure in comparison to the real data) • visualization of the combined signals from the hot regions • Kullback–Leibler(KL) divergences (a measure of the parameter-by-parameter information gain of the posterior over the prior) • background-marginalized likelihood functions(useful in combination with evidence to assess whether additional model complexity is helpful) • model tractability (posterior computational accuracy being higher for less-complex models) • and cross-checking of the inferred background against earlier analysis of PSR J0030+0451 with XMM-Newton. October 5, 2020 – Ecogia Science Meetings 2020–2021 13
Measuring the neutron star equation of state with NICER. Results Riley et al. (2019) Shape • They began with the simplest model, with single-temperature circular spots. • Having the spots be antipodal and identical was quickly ruled out due to large residuals between model and data. • hot region consisted of a circular spot—a core—and a surrounding annulus with an independently determined temperature → too complex • based on different contribution, the model was limited to a single temperature circular spot and a separated annulus • the annulus was tested for an off-centered non-emitting core, and a crescent. • the final best model has a 1T circular spot and and 1T crescent with the same temperature October 5, 2020 – Ecogia Science Meetings 2020–2021 14
Measuring the neutron star equation of state with NICER. Results Riley et al. (2019) Nesting models • The nesting relashionship avoids exploring useless configurations • it provides the simplest allowed model October 5, 2020 – Ecogia Science Meetings 2020–2021 15
Measuring the neutron star equation of state with NICER. Results Riley et al. (2019) Modeling pulses October 5, 2020 – Ecogia Science Meetings 2020–2021 16
Measuring the neutron star equation of state with NICER.
Results Riley et al. (2019)
Mass and radius constraints
• Using the best available geometry, it is possible to constrain mass and
radius independently
• M = 1.34+0.16
−0.15 M and R = 12.71+1.19
−1.14 km, while the compactness
GM/Req c 2 = 0.156+0.008
−0.010 is more tightly constrained
October 5, 2020 – Ecogia Science Meetings 2020–2021 17Measuring the neutron star equation of state with NICER. Results from Miller et al. (2019) Model construction and setup • Hydrogen atmosphere • increasing number of oval spots that can overlap and with different temperature • Multinest runs with 1000 active points • they remove channels below 40 and use best available effective area, but Riley et al. (2019) use the parameterization below with uncertainty of 20% October 5, 2020 – Ecogia Science Meetings 2020–2021 18
Measuring the neutron star equation of state with NICER.
Results from Miller et al. (2019)
Results
• The best configuration has an almost negligible contribution of the small
hot spot at the pole, which has a higher temperature.
• Req = 13.02+1.24 +0.15
−1.06 km and M = 1.44−0.14 M (68%)
October 5, 2020 – Ecogia Science Meetings 2020–2021 19Measuring the neutron star equation of state with NICER. Implications for the equation of state A Bayesian transformation • Posterior distribution of M and R needs to be translated into a relation between pressure and density P(ρ) = K (ρ/ρS )Γ • they use two parametrizations of the EoS: piecewise-polytropic (PP) and speed-of-sound (CS) • To get mass and radius, one needs the EoS and a central density, or equivalently central P and ρ October 5, 2020 – Ecogia Science Meetings 2020–2021 20
Measuring the neutron star equation of state with NICER. Implications for the equation of state Not much gain, for now • Despite the exceptional effort, the updated constraints on EoS from this data set is very limited, many more objects would be needed. • observation of high-mass (2M ) pulsars and chiral effective field theory (??) set tighter priors. October 5, 2020 – Ecogia Science Meetings 2020–2021 21
Measuring the neutron star equation of state with NICER. Implication for NS physics A surprise ! • The most probable hot-spot configuration is completely surprising, but very well constrained. October 5, 2020 – Ecogia Science Meetings 2020–2021 22
Measuring the neutron star equation of state with NICER.
Implication for NS physics
Considerations
• Multipolar fields and non-dipolar configurations were already present in
literature for both rotation-powered and accretion powered pulsars.
• it was clear that there is no symmetry between poles since decades.
• is the polar cap configuration stable ? (we know of pulse switches in
different states, but with repetitive patterns)
• simplifying assumptions? for example:
• atmosphere chemical composition and ionization degree,
• the choice to neglect smooth temperature gradients across the hot regions,
• the consideration of a specific set of hot-region shapes (which can be too
general, or, on the contrary, too specific given their phenomenological
nature),
• the background treatment.
• It will also be interesting to see whether similar results for the other
pulse-profile modeling sources targeted by NICER.
• implication for radio and gamma-ray pulse formation are very promising
• a challenge is finding physically motivated configurations
October 5, 2020 – Ecogia Science Meetings 2020–2021 23Measuring the neutron star equation of state with NICER. Implication for NS physics Future • It would be nice to have such a great constraint, but we will need even larger aperture telescopes October 5, 2020 – Ecogia Science Meetings 2020–2021 24
Measuring the neutron star equation of state with NICER. Implication for NS physics References I Bilous, A. V., Watts, A. L., Harding, A. K., et al. 2019, Astrophys. J., 887, L23, doi: 10.3847/2041-8213/ab53e7 Bogdanov, S., Lamb, F. K., Mahmoodifar, S., et al. 2019a, Astrophys. J., 887, L26, doi: 10.3847/2041-8213/ab5968 Bogdanov, S., Guillot, S., Ray, P. S., et al. 2019b, Astrophys. J., 887, L25, doi: 10.3847/2041-8213/ab53eb Cottam, J., Paerels, F., & Mendez, M. 2002, \nat, 420, 51, doi: 10.1038/nature01159 Miller, M. C., Lamb, F. K., Dittmann, A. J., et al. 2019, Astrophys. J., 887, L24, doi: 10.3847/2041-8213/ab50c5 Raaijmakers, G., Riley, T. E., Watts, A. L., et al. 2019, Astrophys. J., 887, L22, doi: 10.3847/2041-8213/ab451a Riley, T. E., Watts, A. L., Bogdanov, S., et al. 2019, Astrophys. J., 887, L21, doi: 10.3847/2041-8213/ab481c Watts, A. L., Andersson, N., Chakrabarty, D., et al. 2016, Rev. Mod. Phys., 88, 21001. http://link.aps.org/doi/10.1103/RevModPhys.88.021001papers2: //publication/doi/10.1103/RevModPhys.88.021001 October 5, 2020 – Ecogia Science Meetings 2020–2021 25
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