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Quadrupole moments of neutron stars and strange stars. Martin Urbanec , John C. Miller, Zdenek Stuchlk Institute of Physics, Silesian University in Opava , Czech Republic Department of Physics (Astrophysics), University of Oxford, UK. Slow rotation approximation. - PowerPoint PPT Presentation

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Testovn stavovch rovnic pomoc astronomickch pozorovn

Quadrupole moments of neutron stars and strange stars Martin Urbanec, John C. Miller, Zdenek StuchlkInstitute of Physics, Silesian University in Opava, Czech Republic Department of Physics (Astrophysics), University of Oxford, UK

Slow rotation approximationHartle (1967) & Hartle Thorne (1968)Chandrasekhar, Miller (1974) & Miller (1977)Slow rotation approximation:M= 1.4 M R= 12km fmax 1250 HzFastest observed pulsar f=716Hz PSR J1748-2446ad11 pulsars f > 500HzSlow rotation perturbation of spherical symmetryTerms up to 2nd order in are taken into accountRigid rotation

Description of matterRelatively broad set of equation of state is selectedSome of them do not meet requirement of new observational test - Steiner, Lattimer, Brown (2010), Demorest et al. (2010), Podsiadlowski et al. (2005) For strange stars: Simplest MIT bag model is used with two values of bag constant

B=(2x)1014 g.cm-3C=0.15Non-rotating star

Spherically symmetric star Solve equation of hydrostatic equilibrium for given central parameters and using assummed equation of state

Mass Radius relation

Mass Compactness relation

Hartle- Thorne approximationHartle Thorne metric

(r) 1st order in h0(r), h2(r), m0(r), m2(r), k2(r) 2nd order in , functions of r only Put metric into Einstein equations (energy momentum of perfect fluid, or vacuum)Rotating neutron stars key quantitiesWithin the slow rotation approximation only quantities up to 2nd order in are taken into accountM mass of the rotating object J angular momentum Q quadrupole momentThese are defined from the behaviour of the gravitational field at the infinity

Calculation of angular momentum JFrom (t ) component of Einstein equation

Equation is solved with proper boundary condition We want to calculate models for given - rescaling

Calculation of massCalculation of the spherical perturbation (l=0) quantities

Total gravitational mass of the rotating star

Calculation of quadrupole momentCalculation of the deviation from spherical symmetry

where, K comes from matching of internal and external solutionsDescription of rotating starsPhysical properties, that fully describe rotating compact stars within the HT approx. are M, J, QSometimes useful to define dimensionlessj=J/M2q=Q/M3 and frequency independent quantitesmoment of inertia factor I/MR2Kerr parameter QM/J2and express them as a functions of compactness R/2MI/MR2 R/2M relation

R/2M- j relation for 300Hz

QM/J2 R/2M relation

Neutron to strange star transitionAccording to some theories, strange matter could be the most stable form of matterWe do not see it on Earth long relaxation time?Compact stars two possible scenarios of transition (collapse)central pressure overcome critical value (e.g. during accretion)neutron star is hit by strangelet travelling in the Universe

Assume anglar momentum and number of particles being conserved during transitionMass Baryon number

Moment of inertia Baryon number

ConclusionsI/MR2 and QM/J2 could be approximated by analytical function, that hold for all EoS of NS and significantly differs from the one for strange starsAs one goes with R/2M to 1, Kerr parameter goes to Kerr valueNeutron to strange star transition could lead to spin-down of the object (depending on EoS, but more likely for more massive objects)