Relativistic precession of orbits around neutron stars George Pappas SFB video seminar November 15, 2012
Outline 1 /19 Spacetime geometry around neutron stars. Numerical rotating neutron stars with realistic EOSs An analytic spacetime for the exterior of NSs Geodesics and relativistic precession frequencies. Kerr frequencies. Neutron star frequencies. Relation to QPOs. Neutron star properties from frequencies.
Spacetime geometry around neutron stars: Numerical rotating neutron stars 2 /19 The line element for a stationary and axially symmetric spacetime (the spacetime admits a timelike, ξ a, and a spacelike, η a, killing field, i.e. it has rotational symmetry and symmetry in translations in time) is 1, ds 2 = e 2ν dt 2 + r 2 sin 2 θb 2 e 2ν (dφ ωdt) 2 + e 2(ζ ν) (dr 2 + r 2 dθ 2 ), Field equations in the frame of the ZAMOs: D (BDν) = 1 [ (ɛ + p)(1 + u 2 r2 sin 2 θb 3 e 4ν Dω Dω + 4πBe 2ζ 2ν 2 ) + 2p 1 u 2 ], D (r 2 sin 2 θb 3 e 4ν Dω) = 16πr sin θb 2 e 2ζ 4ν (ɛ + p)u 1 u, 2 D (r sin θdb) = 16πr sin θbe 2ζ 2ν p, Komatsu, Eriguchi, and Hechisu 2 proposed a scheme for integrating the field equations using Green s functions. This scheme is implemented by the RNS numerical code to calculate rotating neutron stars 3. 1 E. M. Butterworth and J. R. Ipser, ApJ 204, 200 (1976). 2 H. Komatsu, Y. Eriguchi, and I. Hechisu, MNRAS 237, 355 (1989). 3 N. Stergioulas, J.L. Friedman, ApJ, 444, 306 (1995).
Spacetime geometry around neutron stars: Numerical rotating neutron stars 3 /19 One can use RNS to calculate models of rotating neutron stars for a given equation of state. For example we show here some models for the APR EOS: 2.5 2.5 2 2 1.5 1.5 M Mo M Mo 1 1 0.5 0.5 5 15 20 25 Ρcx 14 12 14 16 18 20 Re The models with the fastest rotation have a spin parameter, j = J/M 2, around 0.7 and a ratio of the polar radius over the equatorial radius, rp/re, around 0.56. The code, except from the various physical characteristics of the neutron stars, provides the metric functions in a grid on the coordinates x and µ in the whole space (for values from 0 to 1 for both variables), where µ = cosθ, r = xr e 1 x and r e is a length scale. The RNS code also calculates the first non-zero multipole moments, i.e., M, J, M2, and S3 4. 4 G.P. and T. A. Apostolatos, Phys. Rev. Lett. 8 (2012), 2314.
Spacetime geometry around neutron stars: An analytic spacetime 4 /19 The vacuum region of a stationary and axially symmetric space-time can be described by the Papapetrou line element 5, ds 2 = f (dt wdφ) 2 + f 1 [ e 2γ ( dρ 2 + dz 2) + ρ 2 dφ 2], where f, w, and γ are functions of the Weyl-Papapetrou coordinates (ρ, z). By introducing the complex potential E(ρ, z) = f(ρ, z) + ıψ(ρ, z) 6, the Einstein field equations take the form, (Re(E)) 2 E = E E, where, f = ξ a ξa and ψ is defined by, aψ = εabcd ξ b c ξ d. An algorithm for generating solutions of the Ernst equation was developed by Sibgatullin and Manko 7. A solution is constructed from a choice of the Ernst potential along the axis of symmetry in the form of a rational function E(ρ = 0, z) = e(z) = P (z) R(z), where P (z), R(z) are polynomials of z of order n with complex coefficients in general. 5 A. Papapetrou, Ann. Phys., 12, 309 (1953). 6 F.J. Ernst, Phys. Rev., 167, 1175 (1968); Phys. Rev., 168, 1415 (1968). 7 V.S. Manko, N.R. Sibgatullin, Class. Quantum Grav.,, 1383 (1993).
Spacetime geometry around neutron stars: An analytic spacetime 5 /19 Two-Soliton spacetime: This is a 4-parameter analytic spacetime which can be produced if one chooses the Ernst potential on the axis to have the form: e(z) = (z M ia)(z + ib) k (z + M ia)(z + ib) k The parameters a, b, k of the spacetime can be related to the first non-zero multipole moments through the equations, where M is the mass. J = am, M2 = (a 2 k)m, S3 = [ a 3 (2a b)k ] M, One can use the multipole moments M, J, M2, and S3 of a numerically calculated neutron star and produce an analytic two-soliton spacetime that reproduces very accurately the numerically calculated spacetime around the neutron star, as well as the relevant properties of the geodesics in that spacetime, such as the various orbital and precession frequencies, the position of the ISCO and the energetics of the orbits 8. 8 G. P. and T. A. Apostolatos, arxiv:1209.6148 [gr-qc]
Geodesics and relativistic precession frequencies 6 /19 Particle motion in a spacetime with symmetries: The symmetry in time translations implies the existence of a timelike Killing vector ξ a and the symmetry in rotations implies the existence of a spacelike Killing vector η a. Symmetry in time translations is associated to an integral of motion, energy E ( ) E = paξ a = pt = gttp t gtφp φ dt = m gtt dτ g dφ tφ dτ (1) Symmetry in rotations is again associated to an integral of motion, angular momentum L L = paη a = pφ = gtφp t + gφφp φ = m ( gtφ dt dτ + g φφ dφ dτ ) (2) From the measure of the four-momentum, p a pa = m 2, we have the equation, ( ) 2 ( ) ( ) ( ) 2 ( ) 2 ( ) 2 dt dt dφ dφ dρ dz 1 = gtt + 2gtφ + gφφ + gρρ + gzz (3) dτ dτ dτ dτ dτ dτ
Geodesics and relativistic precession frequencies 7 /19 Circular equatorial orbits: If we define the angular velocity, Ω dφ, for the circular and equatorial orbits dt equation (3) defines the redshift factor, ( ) 2 dτ = gtt 2gtφΩ gφφω 2, (4) dt and the energy and the angular momentum for the circular orbits take the form, Ẽ E/m = gtt gtφω g tt 2gtφΩ gφφω 2, L L/m = gtφ + gφφω g tt 2gtφΩ gφφω 2. (5) From the conditions, dρ dt = 0, d2 ρ = 0 and dz = 0, and the equations of motion obtained assuming the Lagrangian, L = 1 2 g abẋ a ẋ b, the angular velocity can be calculated dt 2 dt to be, Ω = g tφ,ρ + (gtφ,ρ) 2 gtt,ρgφφ,ρ gφφ,ρ. (6) This is the orbital frequency of a particle in a circular orbit on the equatorial plane.
Geodesics and relativistic precession frequencies 8 /19 More general orbits: Equation (3) can take a more general form in terms of the constants of motion, ( ) 2 ( ) 2 dρ dz gρρ gzz = 1 Ẽ2 gφφ + 2Ẽ Lgtφ + L 2 gtt = Veff, (7) dτ dτ (gtφ) 2 gttgφφ With equation (7) we can study the general properties of the motion of a particle from the properties of the effective potential. Small perturbations from circular equatorial orbits: If we assume small deviations from the circular equatorial orbits of the form, ρ = ρc + δρ and z = δz, then we obtain the perturbed form of (7), gρρ ( d(δρ) dτ ) 2 gzz ( d(δz) dτ ) 2 = 1 2 2 Veff ρ 2 (δρ)2 + 1 2 2 Veff z 2 (δz)2, This equation describes two harmonic oscillators with frequencies,, κ 2 ρ = gρρ 2 2 Veff ρ 2 c, κ 2 z = gzz 2 2 Veff z 2 The differences of these frequencies (corrected with the redshift factor (4)) from the orbital frequency, Ωa = Ω κa, define the precession frequencies. c
Kerr frequencies 9 /19 Frequencies for the Kerr spacetime: Orbital frequency (in Hz): Ω = 299790 M/r 3 1+j(M/r) 3/2 Radial frequency (in Hz): κρ = Ω ( 1 6(M/r) + 8j(M/r) 3/2 3j 2 (M/r) 2) 1/2 Vertical frequency (in Hz): κz = Ω ( 1 4j(M/r) 3/2 + 3j 2 (M/r) 2) 1/2 Orbital, Ω, and precession, Ωa = Ω κa, frequencies for an M = 1.4M Kerr black hole for various j (0.01-0.91). The plots show the frequencies (ν = Ω/2π), 1.4MO Kerr black hole 1.4MO Kerr black hole v Hz vz, vr Hz 0.00001 0.00001 20 50 0 200 500 2000 Rcirc km Plots of the orbital, periastron and nodal precession frequencies. 1 0 vφ Hz Plots of the periastron and the nodal precession frequencies against the orbital frequency.
Kerr frequencies /19 Scaled Frequencies for the Kerr spacetime: Scaled Orbital frequency: MΩ = 299790 1/r 3 1+j(1/r) 3/2 Scaled Radial frequency: Mκρ = MΩ ( 1 6(1/r) + 8j(1/r) 3/2 3j 2 (1/r) 2) 1/2 Scaled Vertical frequency: Mκz = MΩ ( 1 4j(1/r) 3/2 + 3j 2 (1/r) 2) 1/2 Orbital, M νφ, and precession, M νa, scaled frequencies for Kerr black holes for various j (0.01-0.91). Kerr black hole Kerr black hole M vr, M vz km Hz 0.00001 0.00001 5 50 0 500 Plots of the orbital, periastron and nodal precession scaled frequencies. 1 0 0 M vφ km Hz Plots of the periastron and the nodal precession scaled frequencies against the orbital scaled frequency. The general effect of rotation is to increase the observed frequencies (and reduce the ISCO radius).
Neutron star frequencies 11 /19 Frequencies for the spacetime around neutron stars: Orbital frequency: Ω = g tφ,ρ+ (gtφ,ρ) 2 gtt,ρgφφ,ρ Radial frequency: ( { ( ) gρρ 2 (gtt + gtφω) 2 gφφ ρ 2 Vertical frequency: ( { κρ = κz = gzz 2 (gtt + gtφω) 2 ( gφφ ρ 2 ) gφφ,ρ,ρρ 2(g tt + gtφω)(gtφ + gφφω) ( gtφ ρ 2 ),zz 2(g tt + gtφω)(gtφ + gφφω) ( gtφ ρ 2 ) ( + (g tφ + gφφω) 2 gtt,ρρ ρ ),ρρ 2 ( + (g tφ + gφφω) 2 gtt,zz ρ ),zz 2 }) 1/2 }) 1/2 Orbital, M νφ, and precession, M νa, scaled frequencies for neutron star models (132) constructed with the APR EOS for various j up to the Kepler limit (0-0.7) and masses from 1M up to 2.7M. M vr,m vz km Hz 5 20 50 0 200 Plots of the orbital, periastron and nodal precession frequencies. 50 0 500 5000 M vφ km Hz Plots of the periastron and the nodal precession frequencies against the orbital frequency.
Neutron star frequencies 12 /19 Frequencies for the spacetime around neutron stars: The effect of rotation Orbital frequency, periastron precession frequency and nodal precession frequency: Orbital, M νφ, and precession, M νa, scaled frequencies for neutron star models constructed with the APR EOS for various j up to the Kepler limit(0-0.7) and the same central density. Ρc 6.31 14, M 0.924Mo,1.13Mo, j 0.,0.66 Ρc 7.36 14, M 1.2Mo,1.5Mo, j 0.,0.668 1.5 2 2.5 0.5 1 0 12 14 16 18 20 5 20 50 0 200 Ρc. 14, M 1.78Mo,2.22Mo, j 0.,0.682 5 50 0 500 5 20 50 0 200 Plots of the orbital, periastron and nodal precession frequencies for different rotations for models with ρc = 6.3 14 g/cm 3. Same plots for ρc = 7.3 14 g/cm 3 (upper) and ρc = 15 g/cm 3 (lower).
Neutron star frequencies 13 /19 Frequencies for the spacetime around neutron stars: The effect of rotation Orbital frequency, periastron precession frequency and nodal precession frequency: Orbital, M νφ, and precession, M νa, scaled frequencies for neutron star models constructed with the APR EOS for various j up to the Kepler limit(0-0.7). 1.5 2 2.5 0.5 1 0 12 14 16 18 20 5 20 50 0 200 5 50 0 500 5 20 50 0 200 Plots of the orbital, periastron and nodal precession frequencies for fastest (upper) and the slowest (lower) rotating models. As in the case of the Kerr spacetime, the general effect of rotation is to shift up the frequencies (and reduce the ISCO radius).
Neutron star frequencies 14 /19 Frequencies for the spacetime around neutron stars: Effect of higher moments Orbital frequency, periastron precession frequency and nodal precession frequency: Orbital, M νφ, and precession, M νa, scaled frequencies for neutron star models constructed with the APR EOS for approximately the same j = 0.22 and different higher moments. 1.5 2 2.5 0.5 1 0 12 14 16 18 20 5 50 0 500 5 20 50 0 200 M 0.9432 1.082 1.23 1.381 1.531 1.677 1.812 1.935 2.277 j 0.2175 0.2161 0.2152 0.2149 0.2152 0.2162 0.2177 0.2197 0.2309 q -0.5062-0.4016-0.3207-0.2593-0.2133-787 -529-338 -0.0955 j 2-0.0473-0.0467-0.0463-0.0462-0.0463-0.0467-0.0474-0.0483-0.0533 q is the reduced quadrupole, q = M2/M 3, which for the Kerr spacetime is q = j 2.
Neutron star frequencies 15 /19 Frequencies for the spacetime around neutron stars: Effect of higher moments Orbital frequency, periastron precession frequency and nodal precession frequency: Orbital, M νφ, and precession, M νa, scaled frequencies for neutron star models constructed with the APR EOS for approximately the same j = 0.39 and different higher moments. 1.5 2 2.5 0.5 1 0 12 14 16 18 20 5 50 0 500 5 20 50 0 200 M 0.986 1.133 1.288 1.447 1.604 1.755 1.896 2.021 2.368 j 0.3837 0.382 0.381 0.3809 0.3817 0.3834 0.3861 0.3894 0.4069 q -1.427-1.132-0.905-0.7334-0.6044-0.508-0.4362-0.3836-0.2781 j 2-472 -459-452 -451-457 -47-49 -516-655
Neutron star frequencies 16 /19 Frequencies for the spacetime around neutron stars: Effect of higher moments Orbital frequency, periastron precession frequency and nodal precession frequency: Orbital, M νφ, and precession, M νa, scaled frequencies for neutron star models constructed with the APR EOS for approximately the same j = 0.6 and different higher moments. 1.5 2 2.5 0.5 1 0 12 14 16 18 20 5 50 0 500 5 20 50 0 200 M 1.084 1.252 1.429 1.608 1.784 1.951 2.4 2.24 2.599 j 0.5923 0.5932 0.5942 0.5957 0.5979 0.6007 0.6041 0.6082 0.6265 q -2.82-2.242-1.799-1.464-1.215-1.029-0.892-0.7918-0.5912 j 2-0.3508-0.3518-0.3531-0.3549-0.3574-0.3608-0.365-0.3699-0.3925 The general effect of a quadrupole being higher than that of Kerr is to make κz higher than Ω in the region near the ISCO. That region increases with q.
Neutron star frequencies 17 /19 Frequencies for the spacetime around neutron stars: Effect of the EOS 9 Plots of the precession frequencies against the orbital frequency for 3 NS sequences of various rotations for AU, FPS and L EOS. The models of the sequences have masses equal to 1.4M, the mass of the maximum mass non-rotating model that the EOS admits and the mass of a hyper-massive model without non-rotating limit. c b 0 0 1 1 vρ,vz Hz vρ,vz Hz 0.01 0 150 200 300 500 700 1500 2000 0.01 0 150 200 300 500 700 1500 2000 vu Hz vu Hz a 0 1 The general effect of the EOS is that stiffer EOSs produce higher quadrupole. The Kerr spacetime behaves like a super-stiff star. vρ,vz Hz 0.01 0 150 200 300 500 700 1500 2000 vu Hz 9 G.P., 2012 MNRAS, 422, 2581-2589.
Relation to QPOs 18 /19 The orbital frequency, the periastron precession and the nodal precession frequencies are relevant for QPOs from accretion discs, Mechanisms for producing QPOs 11 Typical X-Ray spectrum 12 Apart from orbiting hot spots and oscillations on the disc, one could also have precessing rings or misaligned precessing discs, which either themselves have a modulated emission or they are eclipsing the emission from the central object. Stella & Vietry, 1998, ApJ, 492, L59. 11 F.K. Lamb, Advances in Space Research, 8 (1988) 421. 12 Boutloukos et al., 2006, ApJ, 653, 1435-1444.
Neutron star properties from frequencies. 19 /19 The characteristics of the orbits in a spacetime can be associated to its multipole moments. The precession frequencies are related to the multipole moments, ) Ωρ Ω = 3υ2 4 S 1 + M 2υ3 ( 9 2 3M 2 2M 3 Ωz Ω = 2 S 1 + 3M 2 + M 2υ3 2M 3υ4 ( υ 4 S 1 + M 2υ5 7 S2 1 M + 3 M 2 4 M 3 ) υ 6 + ( 27 2 2 S2 1 M 21M 2 4 2M 3 ( 11 S 1M2 M 6 S 3 5 M 4 ) ) υ 6 +... υ 7 +... where υ = (MΩ) 1/3 and S1 = J. If specific QPOs are associated to the orbital frequencies, they could then be used to measure the multipole moments of the central object 13 and from the moments, probe its structure. 13 G.P., 2012 MNRAS, 422, 2581-2589.