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Propagation of Light About Rapidly Rotating Neutron Stars Sheldon Campbell University of Alberta Motivation • Telescopes are now precise enough to detect thermal spectra from compact stars. • What flux is measured by an observer looking at a rapidly rotating relativistic star? • How accurate are measurements of neutron star radii, estimated from measurements of flux and distance? General Relativistic Effects • gravitational redshift • bending of light – the rear of the star becomes partially visible. • doppler effects due to the star’s rotation – one side of the star is blue-shifted while the other side is red-shifted. • frame-dragging – zero angular momentum orbits have nonzero angular deflection. To discuss flux received by an observer, we must know: • from where on the star each photon originated • the red/blue shift of each photon Ray-tracing must be implemented to accomplish this. Rotating Neutron Star Metric • Spherical coordinates (r, θ, φ) • r corresponds to the isotropic radial coordinate. 2ν 2α ds = − e dt + e ( dr + r d θ ) 2 2 2 2 2 + e 2 β r 2 sin 2 θ ( d φ − ω dt ) 2 • Metric potentials ν, α, β, and ω are functions of r and θ only. Rigid Rotator of Perfect Fluid • 4-velocity of rotator with angular velocity Ω −ν µ e µ µ u = (t + Ωφ ) 1 − e β −ν (Ω − ω ) 2 r 2 sin 2 θ • Perfect fluid Stress-Energy Tensor µν µ ν µν T = (ε + p )u u + pg • Relationship ε(p) is given by EOS • Many candidate equations of state exist that conform with current observed data. Solution for Metric • Einstein’s equations are solved numerically to determine the metric potentials. • Sufficient conditions to state a model are – the equation of state, – gravitational mass M, and – frequency of rotation f. Geodesic Equations dt = e − 2ν (1 − ω b) dλ dφ − 2ν −2 β b = ω e (1 − ω b) + e dλ r 2 sin 2 θ 2 2 d r 1 1 dr dr dθ = α , r + P − R − 2α , r + − 2α ,θ dλ 2 r r dλ dλ dλ 2 dθ 1 (α ,θ P − Θ ) − 2α ,θ dθ − 2α ,r + 1 dr dθ 2 = 2 dλ2 r dλ r dλ dλ b2 P = e −2α Q Q = e − 2ν (1 − ω b) 2 − e − 2 β r 2 sin 2 θ 1 − 2α ∂Q 1 − 2α ∂Q R=− e Θ=− e 2 ∂R 2 ∂θ Initial Conditions • Initial position: to , ro ,θ o , φo • Initial velocity: & ro ,θ o , and the sign of φ o & & Numerical Geodesic Solutions • Use 5th order Runge-Kutta Solver with variable step size. • Accuracy of solutions tested by: – 4-momentum conservation requires 2 2 dr dθ + r2 −P=0 dλ dλ which I use as an independent error estimator, – comparison with 3rd and 4th order solvers, – verify Schwarzschild geodesics when Ω=0. Geodesics from the Star’s Surface φ =0 π 0 ≤θ ≤ 2 rp •Surface parametrized rs(θ) 1 drs re •Let K (θ ) ≡ − rs dθ Light Emitted from Surface Altitude-Azimuth System & cos h sin A φ o = e − ( β +ν ) r sin θ D 1 − cos 2 h sin 2 A ro = ε e − (α +ν ) & 1+ D2 1 − e 2 ( β −ν )ω 2 r 2 sin 2 θ ε 1 1 − cos 2 h sin 2 A θ&o = e − (α +ν ) r 1+ D2 1 − e 2 ( β −ν )ω 2 r 2 sin 2 θ 2 K − ε N (1 + K )ξ 1 − ξ 2 ε N = sign (cos A) D= ξ 2 (1 + K 2 ) − 1 sign ( D + K ), D + K ≠ 0 sin h ε = ξ= ε N , D+K =0 1 − cos 2 h sin 2 A The Star’s Sky Observer θS ∆φ S When a photon’s destination is the same latitude as the observer, a photon originating at the same latitude, altitude and azimuth, at a longitude equal to ∆φ S will reach the observer. Specific Intensity I. o ν The energy of radiation emission • with frequency between νo and νo+dνo • passing through an area dA with normal n ˆ • in time dt • emitted within a solid angle dΩ about the ˆ direction k is ˆ ˆ ˆ dE = Iν o (k , x, t ) k ⋅ n dA dΩ dν o dt Photon Number ˆ, x , t ) = h ν n ( x , p , t ) 4 3 r o r r I vo ( k 2 c Iν o Since n is an invariant, 3 is an invariant. νo We know the photon number for blackbody emission, giving the specific intensity at the surface of the star: 2h νo 3 Iν o = 2 hν o c e kT − 1 Flux from a Star at the Observer 4 2h ν ∞ sin h ν 0 3 dF = 2 dν o dA c ν o d e kT − 1 2 hν o The shift in frequency is given by 2ν 2β ν∞ e − e (Ω − ω ) r sin θ 2 2 2 = . νo 1 − Ωb The area element is α +β dA = e 1 + K r sin θ dθ dφ . 2 2 2 The Distance Seen Behind Some Stars 140 1.400 MSUN, eos A, rotating away Extremal Longitudes (degrees) 135 130 1.400 MSUN, eos A, rotating toward 125 1.400 MSUN, eos L, 120 rotating away 115 1.400 MSUN, eos L, rotating toward 110 1.657 MSUN, eos L, 105 rotating away 100 1.657 MSUN, eos L, 95 rotating toward 2.000 MSUN, eos L, 90 rotating away 0 200 400 600 800 1000 2.000 MSUN, eos L, Frequency of Rotation (Hz) rotating toward Percent Increase in Flux as Rotation Increases 50 Percent Increase in Flux from 40 Spherical Star 1.400 MSUN, eosA 1.400 MSUN, eosL 30 1.657 MSUN, eosL 20 2.000 MSUN, eosL 10 0 0 200 400 600 800 1000 Frequency of Rotation (Hz) Percent Increase of Flux per Area as Rotation Increases Percent Increase of Flux per 10 Area from Spherical Star 8 1.400 MSUN, eosA 6 1.400 MSUN, eosL 1.657 MSUN, eosL 4 2.000 MSUN, eosL 2 0 0 200 400 600 800 1000 Frequency of Rotation (Hz) Determining the Radius of a Star from its Flux • A star’s luminosity radius is defined as 2 Fd RL = . σT 4 • In the Schwarzschild metric, the flux is redshifted. In this case the luminosity radius, when the flux is viewed from infinity, is R R∞ = L . 2M 1− R∞ • Solving for R∞ gives the estimated radius. RL 2 R∞ = M 1 + 1 + 2 M • How does this value compare with the actual equatorial and polar radii of rapidly rotating neutron stars? Comparison of Observed Radii with Actual Radii 21 R_e, 1.4 MSUN, eos A R_p, 1.4 MSUN, eos A 19 Stellar Radius (km) R_inf, 1.4 MSUN, eos A R_e, 1.4 MSUN, eos L 17 R_p, 1.4 MSUN, eos L R_inf, 1.4 MSUN, eos L 15 R_e, 1.657 MSUN, eos L R_p, 1.657 MSUN, eos L 13 R_inf, 1.657 MSUN, eos L R_e, 2.000 MSUN, eos L R_p, 2.000 MSUN, eos L 11 R_inf, 2.000 MSUN, eos L 9 0 200 400 600 800 1000 Rotation Frequency (Hz) Percent Difference Between R_e and R_inf 18 16 14 Percent Difference 12 1.400 MSUN, eos A 10 1.400 MSUN, eos L 1.657 MSUN, eos L 8 2.000 MSUN, eos L 6 4 2 0 0 200 400 600 800 1000 Frequency of Rotation (Hz) Future Direction for this Research • Implement observers at any angle in the star’s sky. • Calculate the stellar spectrum received by the observer. • Consider the rotational broadening of spectral lines. I would like to thank: • Sharon Morsink for introducing me to this problem and guiding me through it. • NSERC for its support of this research.

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posted: | 11/30/2011 |

language: | English |

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