Constraining the Radius of Neutron Stars Through the Moment of Inertia Neutron star mergers: From gravitational waves to nucleosynthesis International Workshop XLV on Gross Properties of Nuclei and Nuclear Excitations Hirschegg, Kleinwalsertal, Austria, January 15-21, 2017 European Research Council Established by the European Commission Observation Theoretical description of neutron stars Equation of state and neutron star structure Constraining the radius of neutron stars Summary
▶ knowledge of EOS is restricted ▶ relationship between EOS and MR relation ▶ precise mass measurements are possible ▶ radius determination influenced by systematic uncertainties ▶ moment of inertia measurement seems feasible in the future ►  PSR J0737–3039A, M = 1.337(5) M o  Burgay et al., Nature (2003); Lyne et al., Science (2004) ▶ goal: constraining radius/EOS by moment of inertia
Courticsy of J. M. Latims

▶ pulsar observation —→ determine post–Keplerian parameters stars, Living Rev. Relstivity (2003) ▶ post–Keplerian parameters as functions of the masses  Damour and Deruelle, Ann. Inst. Henri Peincare (1988) intersection area yields masses for a binary system see also talk of Paulo Freire Kramar et al., Science (2008)
▶ relativistic spin-orbit (SO) coupling causes an additional contribution to advance of the periastron ▶ spin of PSR J0737–3039B is negligible ▶ contributions to advance of periastron ω = W1pN + W2pN + WSO (pN: post-Newtonian) Kramer & Wex, Class. Quant. Grae. (2009)
▶ relativistic spin-orbit (SO) coupling causes an additional contribution to advance of the periastron ▶ spin of PSR J0737–3039B is negligible ▶ contributions to advance of periastron ω = W1pN + W2pN + WSO (pN: post-Newtonian) Kramer & Wex, Class, Quant, Gray, (2009) ▶ candidate for future measurement: PSR J0737–3039A ▶ highly relativistic binary Kramer & Wax, Class. Quantum Grav. (2009) Theoretical description of neutron stars Non-rotating neutron stars ▶ hydrostatic equilibrium Schwarzschild metric
TOV equations
: 4πcr² dr
2m
January 19, 2017 | Institut für Kemphysäk | Svenja Kim Greäl | 8
Theoretical description of neutron stars Slowly rotating neutron stars ▶ hydrostatic equilibrium ▶ Schwarzschild metric → Hartle-Thorne metric Harte, APJ (1967); Harte & Thorne, APJ (1968) ds² = −e' dt² + èdr² + r²dθ² + r²sin² θ(dφ − (ω + 0 (Ω ▶ TOV equations remain
dm dr du Theoretical description of neutron stars Moment of inertia Hartle, APJ (1967), Hartle & Thorne, APJ (1968) auxiliary function
in addition to TOV eqns. (概= Ω A
moment of inertia
Hebeler, Lattimer, Pethick, Schwenk, Phya, Rey, Lett, (2010); Hebeler, Lattimer, Pethick, Schwenk, APJ (2013) see also taik of Kai Hebele ▶ low density regime: knowledge of nuclear physics ► BPS crust up to psat/2 ▶ chiral EFT expansion up to ～ Psat ▶ high density regime: requirement of causality and constraints from 2.01 Mﻦneutron stars ▶ polytropic expansion Read, Ladey, Owen, Friedman, Phys. Rev. D (2009) er, Latsmer, Pethick, Schwenk, APJ (2013)
hwenk, Ptrya, Rev, Lett, (2010); Hebeler, Lattimer, Pethick, Schwenk, APJ (2013) see also talk of Kei Hebele ▶ low density regime: knowledge of nuclear physics ► BPS crust up to Psat/2 ▶ chiral EFT expansion up to ～ Psat ▶ high density regime: requirement of causality and constraints from 2.01 Mo neutron stars ▶ polytropic expansion Read, Ladey, Owen, Friedman, Phys. Rev. D (2002) or Lattimar Pathick, Schwark, AP,I (2015)
Hebeler, Lattimer, Pethick, Schwenk, Phys. Rev. Lett. (2010); Hebeler, Lattimer, Pethick, Schwenk, APJ (2013) see also taik of Kai Hebeler low density regime: knowledge of nuclear physicsムBPS crust up to psat/2 ▶ chiral EFT expansion up to ～ Psat a high density regime: requirement of causality and constraints from 2.01 Mo neutron stars ▶ polytropic expansion Read, Ladey, Owen, Friedman, Phys. Rev. D (2009) ▶ radius prediction for PSR J0737–3039A: R ≈ (9.9 – 13.6) km Hubeler, Latimer, Pethick, Schwenk, APJ (2013)
grey band holds for 0.07 M ⊙ km -1 ≤ MR -1 several studies have investigated I ( MR −1 )  Beiger & Ha 005): Breu & Rezzolla, MNRAS (2 → dimensionless moment of inertia is not sensitive to EOS A find fit and determine radius constraints
grey band holds for 0.07 Me km-1 < MR-1 several studies have investigated I (MR-1) S): Breu & Rezzolla, MNRAS → dimensionless moment of inertia is not sensitive to EOS ▶ find fit and determine radius constraints ▶ our approach: use the whole EOS uncertainty band created by polytropic expansion
use all EOS from uncertainty band A consider different masses A use I(M) band in order to find A reasonable moment of inertia values
▶ use all EOS from uncertainty band consider different masses A ▶ use I(M) band in order to find reasonable moment of inertia values ▶ combine mass, moment of inertia and Δ /
▶ use all EOS from uncertainty band consider different masses A ▶ use I(M) band in order to find reasonable moment of inertia values ▶ combine mass, moment of inertia and Δ / ▶ assumption for PSR J0737–3039A: l = (70 ± 7) Mckm² R ≈ (9.9 − 13.6) km
▶ use all EOS from uncertainty band ▶ consider different masses ▶ use I(M) band in order to find reasonable moment of inertia values ▶ combine mass, moment of inertia and Δ / ▶ assumption for PSR J0737–3039A: l = (70 ± 7) Mekm² R ≈ (9.9 – 13.6) km 4 R ≈ (11.2 – 12.9) km
▶ derived constraints for the EOS over wide range of densities and resulting NS radii using ▶ BPS crust EOS up to ρεπ/2 ▶ results based on chiral EFT interactions up to ～ Psat ▶ causality at all densities ▶ M max > 2.01 M ● + assumed fixed measured values for moment of inertia (including uncertainties) ▶ developed a framework to perform an extensive large scale sampling of all possible high-density extensions for EOS compatible with constraints ▶ find a reduction of radius uncertainty from moment of inertia measurements by about 50 % (ΔΙ = ±10 %) In collaboration with K. Hebeler and A. Schwenk.
▶ derived constraints for the EOS over wide range of densities and resulting NS radii using ▶ BPS crust EOS up to ρεαι /2 ▷ results based on chiral EFT interactions up to ～ Psat ▶ causality at all densities ▶ M max > 2.01 M ● + assumed fixed measured values for moment of inertia (including uncertainties) ▶ developed a framework to perform an extensive large scale sampling of all possible high-density extensions for EOS compatible with constraints ▶ find a reduction of radius uncertainty from moment of inertia measurements by about 50 % (ΔΙ = ±10 %) In collaboration with K. Hebeler and A. Schwenk.