Quadrupole moments of neutron stars and strange stars

Martin Urbanec, John C. Miller, Zdenek StuchlíkInstitute of Physics, Silesian University in Opava, Czech Republic Department of Physics (Astrophysics), University of Oxford, UK

Slow rotation approximation

• Hartle (1967) & Hartle – Thorne (1968)• Chandrasekhar, Miller (1974) & Miller (1977)• Slow rotation approximation:• M= 1.4 Mʘ R= 12km fmax ≈ 1250 Hz• Fastest observed pulsar – f=716Hz PSR J1748-2446ad• 11 pulsars f > 500Hz• Slow rotation – perturbation of spherical symmetry• Terms up to 2nd order in Ω are taken into account• Rigid rotation

Description of matter

• Relatively broad set of equation of state is selected• Some 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-3

αC=0.15

Non-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 approximation

Hartle – Thorne metric

• ω(r) – 1st order in Ω • h0(r), h2(r), m0(r), m2(r), k2(r) – 2nd order in Ω, functions of r only• Put metric into Einstein equations (energy momentum of perfect fluid, or vacuum)

Rotating neutron stars – key quantities• Within the slow rotation approximation only quantities

up to 2nd order in Ω are taken into account– M … mass of the rotating object– J … angular momentum– Q … quadrupole moment

• These are defined from the behaviour of the gravitational field at the infinity

Calculation of angular momentum J• From (t ϕ) component of Einstein equation

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

Calculation of mass

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

• Total gravitational mass of the rotating star

Calculation of quadrupole moment

• Calculation of the deviation from spherical symmetry

where, K comes from matching of internal and external solutions

Description of rotating stars• Physical properties, that fully describe rotating

compact stars within the HT approx. are M, J, Q• Sometimes useful to define dimensionless– j=J/M2

– q=Q/M3

• and frequency independent quantites– moment of inertia factor I/MR2

– Kerr parameter QM/J2

• and express them as a functions of compactness R/2M

I/MR2 – R/2M relation

R/2M- j relation for 300Hz

QM/J2 – R/2M relation

Neutron to strange star transition

• According to some theories, strange matter could be the most stable form of matter

• We 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 transition

Mass – Baryon number

Moment of inertia – Baryon number

Conclusions

• I/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 stars

• As one goes with R/2M to 1, Kerr parameter goes to Kerr value

• Neutron to strange star transition could lead to spin-down of the object (depending on EoS, but more likely for more massive objects)